{"id":239,"date":"2024-10-18T21:13:19","date_gmt":"2024-10-18T21:13:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/functions-get-stronger\/"},"modified":"2024-10-18T21:13:19","modified_gmt":"2024-10-18T21:13:19","slug":"functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/functions-get-stronger\/","title":{"raw":"Functions: Get Stronger","rendered":"Functions: Get Stronger"},"content":{"raw":"\n

Functions and Function Notation<\/h2>\n

Verbal<\/h4>\n
    \n\t
  1. What is the difference between a relation and a function?<\/li>\n\t
  2. Why does the vertical line test tell us whether the graph of a relation represents a function?<\/li>\n\t
  3. Why does the horizontal line test tell us whether the graph of a function is one-to-one?<\/li>\n<\/ol>\n

    Algebraic<\/h4>\n

    For the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].<\/strong><\/p>\n

      \n\t
    1. [latex]y=x^2[\/latex]<\/li>\n\t
    2. [latex]3x^2+y=14[\/latex]<\/li>\n\t
    3. [latex]y=\u22122x^2+40x[\/latex]<\/li>\n\t
    4. [latex]x=\\frac{3y+5}{7y\u22121}[\/latex]<\/li>\n<\/ol>\n

      For the following exercises, evaluate [latex]f(\u22123),f(2),f(\u2212a),\u2212f(a),f(a+h)[\/latex].<\/strong><\/p>\n

        \n\t
      1. [latex]f(x)=2x\u22125[\/latex]<\/li>\n\t
      2. [latex]f(x)=\\sqrt{2\u2212x}+5[\/latex]<\/li>\n\t
      3. [latex]f(x)=|x\u22121|\u2212|x+1|[\/latex]<\/li>\n<\/ol>\n

        Graphical<\/h4>\n

        For the following exercises, use the vertical line test to determine which graphs show relations that are functions.<\/strong><\/p>\n

          \n\t

        1. \n\"Graph<\/li>\n\t

        2. \n\"Graph<\/li>\n\t

        3. \n\"Graph<\/li>\n<\/ol>\n

          For the following exercises, determine if the given graph is a one-to-one function.<\/strong><\/p>\n

            \n\t

          1. \n\"Graph<\/li>\n\t

          2. \n\"Graph<\/li>\n\t

          3. \n\"Graph<\/li>\n<\/ol>\n

            Numeric<\/h4>\n

            For the following exercises, determine whether the relation represents a function.<\/strong><\/p>\n

              \n\t
            1. [latex]{(3,4),(4,5),(5,6)}[\/latex]<\/li>\n<\/ol>\n

              For the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].<\/strong><\/p>\n

                \n\t

              1. \n\n\n\n\n
                [latex]x[\/latex]<\/strong><\/td>\n5<\/td>\n10<\/td>\n15<\/td>\n<\/tr>\n
                [latex]y[\/latex]<\/strong><\/td>\n3<\/td>\n8<\/td>\n14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n\t

              2. \n\n\n\n\n
                [latex]x[\/latex]<\/strong><\/td>\n5<\/td>\n10<\/td>\n10<\/td>\n<\/tr>\n
                [latex]y[\/latex]<\/strong><\/td>\n3<\/td>\n8<\/td>\n14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                For the following exercise, use the function [latex]f[\/latex] represented in the table below.<\/strong><\/p>\n

                  \n\t
                1. Solve [latex]f(x)=1[\/latex].\n\n\n\n\n\n\n
                  [latex]x[\/latex]<\/strong><\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n<\/tr>\n
                  [latex]f(x)[\/latex]<\/strong><\/td>\n74<\/td>\n28<\/td>\n1<\/td>\n53<\/td>\n56<\/td>\n3<\/td>\n36<\/td>\n45<\/td>\n14<\/td>\n47<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                  For the following exercises, evaluate the function [latex]f[\/latex] at the values [latex]f(\u22122),f(\u22121),f(0),f(1),[\/latex] and [latex]f(2)[\/latex].<\/strong><\/p>\n

                    \n\t
                  1. [latex]f(x)=f(x)=8\u22123x[\/latex]<\/li>\n\t
                  2. [latex]f(x)=3+\\sqrt{x+3}[\/latex]<\/li>\n\t
                  3. [latex]f(x)=3^x[\/latex]<\/li>\n<\/ol>\n

                    For the following exercise, evaluate the expression, given functions [latex]f[\/latex],[latex]g[\/latex], and [latex]h[\/latex]: <\/strong><\/p>\n

                    [latex]f(x)=3x\u22122[\/latex]<\/strong><\/p>\n

                    [latex]g(x)=5\u2212x^2[\/latex]<\/strong><\/p>\n

                    [latex]h(x)=\u22122x^2+3x\u22121[\/latex] <\/strong><\/p>\n

                      \n\t
                    1. [latex]f(\\frac{7}{3})\u2212h(\u22122)[\/latex]<\/li>\n<\/ol>\n

                      For the following exercise, graph [latex]y=x^2[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                        \n\t
                      1. [latex][\u221210,10][\/latex]<\/li>\n<\/ol>\n

                        For the following exercises, graph [latex]y=x^3[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                          \n\t
                        1. [latex][\u22120.1,0.1][\/latex]<\/li>\n\t
                        2. [latex][\u2212100,100][\/latex]<\/li>\n<\/ol>\n

                          For the following exercise, graph [latex]y=\\sqrt{x}[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                            \n\t
                          1. [latex][0,100][\/latex]<\/li>\n<\/ol>\n

                            For the following exercises, graph [latex]y=\\sqrt[3]{x}[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                              \n\t
                            1. [latex][\u22120.001, 0.001][\/latex]<\/li>\n\t
                            2. [latex][\u22121,000,000, 1,000,000][\/latex]<\/li>\n<\/ol>\n

                              Real-World Applications<\/strong><\/p>\n

                                \n\t
                              1. The number of cubic yards of dirt, [latex]D[\/latex], needed to cover a garden with area [latex]a[\/latex] square feet is given by [latex]D=g(a)[\/latex].
                                \n
                                  \n\t
                                1. A garden with area [latex]5000 ft^2 [\/latex] requires [latex]50 yd^3[\/latex] of dirt. Express this information in terms of the function [latex]g[\/latex].<\/li>\n\t
                                2. Explain the meaning of the statement [latex]g(100)=1[\/latex].<\/li>\n<\/ol>\n<\/li>\n\t
                                3. Let [latex]h(t)[\/latex] be the height above ground, in feet, of a rocket [latex]t[\/latex] seconds after launching. Explain the meaning of each statement:
                                  \n
                                    \n\t
                                  1. [latex]h(1)=200[\/latex]<\/li>\n\t
                                  2. [latex]h(2)=350[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                    Linear Functions<\/h2>\n

                                    Verbal<\/h4>\n
                                      \n\t
                                    1. Terry is skiing down a steep hill. Terry's elevation, [latex]E(t)[\/latex], in feet after t seconds is given by [latex]E(t)=3000\u221270t[\/latex]. Write a complete sentence describing Terry\u2019s starting elevation and how it is changing over time.<\/li>\n\t
                                    2. A boat is [latex]100[\/latex] miles away from the marina, sailing directly toward it at [latex]10[\/latex] miles per hour. Write an equation for the distance of the boat from the marina after [latex]t [\/latex]hours.<\/li>\n<\/ol>\n

                                      Algebraic<\/h4>\n

                                      For the following exercises, determine whether the equation of the curve can be written as a linear function.<\/strong><\/p>\n

                                        \n\t
                                      1. [latex]y=3x-5[\/latex]<\/li>\n\t
                                      2. [latex]3x+5y=15[\/latex]<\/li>\n\t
                                      3. [latex]3x+5y^2=15[\/latex]<\/li>\n<\/ol>\n

                                        For the following exercises, determine whether each function is increasing or decreasing.<\/strong><\/p>\n

                                          \n\t
                                        1. [latex]g(x)=5x+6[\/latex]<\/li>\n\t
                                        2. [latex]b(x)=8\u22123x[\/latex]<\/li>\n\t
                                        3. [latex]k(x)=\u22124x+1[\/latex]<\/li>\n<\/ol>\n

                                          For the following exercises, find the slope of the line that passes through the two given points.<\/strong><\/p>\n

                                            \n\t
                                          1. [latex](1,5)[\/latex] and [latex](4,11)[\/latex]<\/li>\n\t
                                          2. [latex](8,\u20132)[\/latex] and [latex](4,6)[\/latex]<\/li>\n<\/ol>\n

                                            For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.<\/strong><\/p>\n

                                              \n\t
                                            1. [latex]f(\u22125)=\u22124[\/latex] and [latex]f(5)=2[\/latex]<\/li>\n\t
                                            2. Passes through [latex](2,4)[\/latex] and [latex](4,10)[\/latex]<\/li>\n\t
                                            3. Passes through [latex](-1,4)[\/latex] and [latex](5,2)[\/latex]<\/li>\n<\/ol>\n

                                              For the following exercises, find the x<\/em>- and y-<\/em>intercepts of each equation.<\/strong><\/p>\n

                                                \n\t
                                              1. [latex]f(x)=\u2212x+2[\/latex]<\/li>\n\t
                                              2. [latex]h(x)=3x\u22125[\/latex]<\/li>\n\t
                                              3. [latex]\u22122x+5y=20[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. <\/strong><\/p>\n

                                                  \n\t
                                                1. Line 1: Passes through [latex](0,6)[\/latex] and [latex](3,\u221224)[\/latex]
                                                  \nLine 2: Passes through [latex](\u22121,19)[\/latex] and [latex](8,\u221271)[\/latex]<\/li>\n\t
                                                2. Line 1: Passes through [latex](2,3)[\/latex] and [latex](4,\u22121)[\/latex]
                                                  \nLine 2: Passes through [latex](6,3)[\/latex] and [latex](8,5)[\/latex]<\/li>\n\t
                                                3. Line 1: Passes through [latex](2,5)[\/latex] and [latex](5,\u22121)[\/latex]
                                                  \nLine 2: Passes through [latex](\u22123,7)[\/latex] and [latex](3,\u22125)[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises, write an equation for the line described.<\/strong><\/p>\n

                                                    \n\t
                                                  1. Write an equation for a line parallel to [latex]g(x)=3x\u22121[\/latex] and passing through the point [latex](4,9)[\/latex].<\/li>\n\t
                                                  2. Write an equation for a line perpendicular to [latex]p(t)=3t+4[\/latex] and passing through the point [latex](3,1)[\/latex].<\/li>\n<\/ol>\n

                                                    Graphical<\/h4>\n

                                                    For the following exercise, find the slope of the line graphed.<\/strong><\/p>\n

                                                      \n\t

                                                    1. \n\"This<\/li>\n<\/ol>\n

                                                      For the following exercises, write an equation for the line graphed.<\/strong><\/p>\n

                                                        \n\t

                                                      1. \n\"Graph<\/li>\n\t

                                                      2. \n\"Graph<\/li>\n\t

                                                      3. \n\"Graph<\/li>\n<\/ol>\n

                                                        For the following exercises, match the given linear equation with its graph in the figure below.<\/strong><\/p>\n

                                                        \"Graph<\/div>\n
                                                          \n\t
                                                        1. [latex]f(x)=\u22123x\u22121[\/latex]<\/li>\n\t
                                                        2. [latex]f(x)=2[\/latex]<\/li>\n\t
                                                        3. [latex]f(x)=3x+2[\/latex]<\/li>\n<\/ol>\n

                                                          For the following exercises, sketch a line with the given features.<\/strong><\/p>\n

                                                            \n\t
                                                          1. An x-intercept [latex](\u20132,0)[\/latex] and y-intercept of [latex](0,4)[\/latex]<\/li>\n\t
                                                          2. A y-intercept of [latex](0,3)[\/latex] and slope [latex]\\frac{2}{5}[\/latex]<\/li>\n\t
                                                          3. Passing through the points [latex](\u20133,\u20134)[\/latex] and [latex](3,0)[\/latex]<\/li>\n<\/ol>\n

                                                            For the following exercises, sketch the graph of each equation.<\/strong><\/p>\n

                                                              \n\t
                                                            1. [latex]f(x)=\u22123x+2[\/latex]<\/li>\n\t
                                                            2. [latex]f(x)=\\frac{2}{3}x\u22123[\/latex]<\/li>\n<\/ol>\n

                                                              For the following exercises, write the equation of the line shown in the graph.<\/strong><\/p>\n

                                                                \n\t

                                                              1. \n\"The<\/li>\n\t

                                                              2. \n\"Graph<\/li>\n<\/ol>\n

                                                                Numeric<\/h4>\n

                                                                For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.<\/strong><\/p>\n

                                                                  \n\t
                                                                1. \n\n\n\n\n
                                                                  [latex]x[\/latex]<\/strong><\/td>\n[latex]0[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n[latex]15[\/latex]<\/td>\n<\/tr>\n
                                                                  [latex]g(x)[\/latex]<\/strong><\/td>\n[latex]5[\/latex]<\/td>\n[latex]-10[\/latex]<\/td>\n[latex]-25[\/latex]<\/td>\n[latex]-40[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n\t
                                                                2. \n\n\n\n\n
                                                                  [latex]x[\/latex]<\/strong><\/td>\n[latex]0[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n[latex]15[\/latex]<\/td>\n<\/tr>\n
                                                                  [latex]f(x)[\/latex]<\/td>\n[latex]-5[\/latex]<\/td>\n[latex]20[\/latex]<\/td>\n[latex]45[\/latex]<\/td>\n[latex]70[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                  Technology<\/h4>\n

                                                                  For the following exercises, use a calculator or graphing technology to complete the task.<\/strong><\/p>\n

                                                                    \n\t
                                                                  1. If [latex]f[\/latex] is a linear function, [latex]f(0.1)=11.5[\/latex], and [latex]f(0.4)=\u20135.9[\/latex], find an equation for the function.<\/li>\n\t
                                                                  2. Graph the function [latex]f[\/latex] on a domain of [latex][\u201310,10]: f(x)=2,500x+4,000[\/latex].<\/li>\n\t
                                                                  3. The table shows the input, [latex]p[\/latex], and output, [latex]q[\/latex], for a linear function [latex]q[\/latex].\n\n\n
                                                                      \n\t
                                                                    1. Fill in the missing values of the table.<\/li>\n\t
                                                                    2. Write the linear function [latex]q[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n\n
                                                                      [latex]p[\/latex]<\/strong><\/td>\n[latex]0.5[\/latex]<\/td>\n[latex]0.8[\/latex]<\/td>\n[latex]12[\/latex]<\/td>\n[latex]b[\/latex]<\/td>\n<\/tr>\n
                                                                      [latex]q[\/latex]<\/strong><\/td>\n[latex]400[\/latex]<\/td>\n[latex]700[\/latex]<\/td>\n[latex]a[\/latex]<\/td>\n[latex]1,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                                                        \n\t
                                                                      1. Graph the linear function [latex]f[\/latex] on a domain of [latex][\u22120.1,0.1][\/latex] for the function whose slope is [latex]75[\/latex] and y-intercept is [latex]\u221222.5[\/latex]. Label the points for the input values of [latex]\u22120.1[\/latex] and [latex]0.1[\/latex].<\/li>\n<\/ol>\n

                                                                        Real-World Applications<\/h4>\n
                                                                          \n\t
                                                                        1. A gym membership with two personal training sessions costs [latex]$125[\/latex], while gym membership with five personal training sessions costs [latex]$260[\/latex]. What is cost per session?<\/li>\n\t
                                                                        2. A phone company charges for service according to the formula: [latex]C(n)=24+0.1n[\/latex], where [latex]n[\/latex] is the number of minutes talked, and [latex]C(n)[\/latex] is the monthly charge, in dollars. Find and interpret the rate of change and initial value.<\/li>\n\t
                                                                        3. A city\u2019s population in the year 1960 was [latex]287,500[\/latex]. In 1989 the population was [latex]275,900[\/latex]. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.<\/li>\n\t
                                                                        4. Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: [latex]I(x)=1054x+23,286[\/latex], where [latex]x[\/latex] is the number of years after 1990. Which of the following interprets the slope in the context of the problem?
                                                                          \n
                                                                            \n\t
                                                                          1. As of 1990, average annual income was [latex]$23,286[\/latex].<\/li>\n\t
                                                                          2. In the ten-year period from 1990\u20131999, average annual income increased by a total of [latex]$1,054[\/latex].<\/li>\n\t
                                                                          3. Each year in the decade of the 1990s, average annual income increased by [latex]$1,054[\/latex].<\/li>\n\t
                                                                          4. Average annual income rose to a level of [latex]$23,286[\/latex] by the end of 1999.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                            Quadratic Functions<\/h2>\n

                                                                            Verbal<\/h4>\n
                                                                              \n\t
                                                                            1. Explain the advantage of writing a quadratic function in standard form.<\/li>\n\t
                                                                            2. Explain why the condition of [latex]a\u22600[\/latex] is imposed in the definition of the quadratic function.<\/li>\n\t
                                                                            3. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/li>\n<\/ol>\n

                                                                              Algebraic<\/h4>\n

                                                                              For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/strong><\/p>\n

                                                                                \n\t
                                                                              1. [latex]g(x)=x^2+2x\u22123[\/latex]<\/li>\n\t
                                                                              2. [latex]f(x)=x^2+5x\u22122[\/latex]<\/li>\n\t
                                                                              3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\n<\/ol>\n

                                                                                For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/strong><\/p>\n

                                                                                  \n\t
                                                                                1. [latex]f(x)=2x^2\u221210x+4[\/latex]<\/li>\n\t
                                                                                2. [latex]f(x)=4x^2+x\u22121[\/latex]<\/li>\n\t
                                                                                3. [latex]f(x)=\\frac{1}{2}x^2+3x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                  For the following exercises, determine the domain and range of the quadratic function.<\/strong><\/p>\n

                                                                                    \n\t
                                                                                  1. [latex]f(x)=(x\u22123)^2+2[\/latex]<\/li>\n\t
                                                                                  2. [latex]f(x)=x^2+6x+4[\/latex]<\/li>\n\t
                                                                                  3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\n<\/ol>\n

                                                                                    For the following exercises, use the vertex [latex](h,k)[\/latex] and a point on the graph [latex](x,y)[\/latex] to find the general form of the equation of the quadratic function.<\/strong><\/p>\n

                                                                                      \n\t
                                                                                    1. [latex](h,k)=(\u22122,\u22121),(x,y)=(\u22124,3)[\/latex]<\/li>\n\t
                                                                                    2. [latex](h,k)=(2,3),(x,y)=(5,12)[\/latex]<\/li>\n\t
                                                                                    3. [latex](h,k)=(3,2),(x,y)=(10,1)[\/latex]<\/li>\n<\/ol>\n

                                                                                      Graphical<\/h4>\n

                                                                                      For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/strong><\/p>\n

                                                                                        \n\t
                                                                                      1. [latex]f(x)=x^2\u22126x\u22121[\/latex]<\/li>\n\t
                                                                                      2. [latex]f(x)=x^2\u22127x+3[\/latex]<\/li>\n\t
                                                                                      3. [latex]f(x)=4x^2\u221212x\u22123[\/latex]<\/li>\n<\/ol>\n

                                                                                        For the following exercises, write the equation for the graphed quadratic function.<\/strong><\/p>\n

                                                                                          \n\t

                                                                                        1. \n\"Graph<\/li>\n\t

                                                                                        2. \n\"Graph
                                                                                          \n
                                                                                          \n<\/li>\n<\/ol>\n

                                                                                          Numeric<\/h4>\n

                                                                                          For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.<\/strong><\/p>\n

                                                                                            \n\t
                                                                                          1. \n\n\n\n\n
                                                                                            [latex]x[\/latex]<\/strong><\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
                                                                                            [latex]y[\/latex]<\/strong><\/td>\n[latex]1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n\t
                                                                                          2. \n\n\n\n\n
                                                                                            [latex]x[\/latex]<\/strong><\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
                                                                                            [latex]y[\/latex]<\/strong><\/td>\n[latex]-8[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                                            Power Functions and Polynomial Functions<\/h2>\n

                                                                                            Verbal<\/h4>\n
                                                                                              \n\t
                                                                                            1. Explain the difference between the coefficient of a power function and its degree.<\/li>\n\t
                                                                                            2. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/li>\n\t
                                                                                            3. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As [latex]x\u2192\u2212\u221e[\/latex], [latex]f(x)\u2192\u2212\u221e[\/latex] and as [latex]x\u2192\u221e[\/latex], [latex]f(x)\u2192\u2212\u221e[\/latex].<\/li>\n<\/ol>\n

                                                                                              Algebraic<\/h4>\n

                                                                                              For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/strong><\/p>\n

                                                                                                \n\t
                                                                                              1. [latex]f(x)=(x^2)^3[\/latex]<\/li>\n\t
                                                                                              2. [latex]f(x)=\\frac{x^2}{x^2\u22121}[\/latex]<\/li>\n\t
                                                                                              3. [latex]f(x)=3^{x+1}[\/latex]<\/li>\n<\/ol>\n

                                                                                                For the following exercises, find the degree and leading coefficient for the given polynomial.<\/strong><\/p>\n

                                                                                                  \n\t
                                                                                                1. [latex]7\u22122x^2[\/latex]<\/li>\n\t
                                                                                                2. [latex]x(4\u2212x^2)(2x+1)[\/latex]<\/li>\n<\/ol>\n

                                                                                                  For the following exercises, determine the end behavior of the functions.<\/strong><\/p>\n

                                                                                                    \n\t
                                                                                                  1. [latex]f(x)=x^4[\/latex]<\/li>\n\t
                                                                                                  2. [latex]f(x)=\u2212x^4[\/latex]<\/li>\n\t
                                                                                                  3. [latex]f(x)=\u22122x^4\u22123x^2+x\u22121[\/latex]<\/li>\n<\/ol>\n

                                                                                                    For the following exercises, find the intercepts of the functions.<\/strong><\/p>\n

                                                                                                      \n\t
                                                                                                    1. [latex]f(t)=2(t\u22121)(t+2)(t\u22123)[\/latex]<\/li>\n\t
                                                                                                    2. [latex]f(x)=x^4\u221216[\/latex]<\/li>\n\t
                                                                                                    3. [latex]f(x)=x(x^2\u22122x\u22128)[\/latex]<\/li>\n<\/ol>\n

                                                                                                      Graphical<\/h4>\n

                                                                                                      For the following exercises, determine the least possible degree of the polynomial function shown.<\/strong><\/p>\n

                                                                                                        \n\t

                                                                                                      1. \n\"Graph<\/li>\n\t

                                                                                                      2. \n\"Graph<\/li>\n\t

                                                                                                      3. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                        For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/strong><\/p>\n

                                                                                                          \n\t

                                                                                                        1. \n\"Graph<\/li>\n\t

                                                                                                        2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                          Numeric<\/h4>\n

                                                                                                          For the following exercises, make a table to confirm the end behavior of the function.<\/strong><\/p>\n

                                                                                                            \n\t
                                                                                                          1. [latex]f(x)=x^4\u22125x^2[\/latex]<\/li>\n\t
                                                                                                          2. [latex]f(x)=(x\u22121)(x\u22122)(3\u2212x)[\/latex]<\/li>\n<\/ol>\n

                                                                                                            Technology<\/h4>\n

                                                                                                            For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/strong><\/p>\n

                                                                                                              \n\t
                                                                                                            1. [latex]f(x)=x^3(x\u22122)[\/latex]<\/li>\n\t
                                                                                                            2. [latex]f(x)=x(14\u22122x)(10\u22122x)[\/latex]<\/li>\n\t
                                                                                                            3. [latex]f(x)=x^3\u221216x[\/latex]<\/li>\n<\/ol>\n

                                                                                                              Graphs of Polynomial Functions<\/h2>\n

                                                                                                              Verbal<\/h4>\n
                                                                                                                \n\t
                                                                                                              1. What is the difference between an [latex]x[\/latex]- intercept and a zero of a polynomial function [latex]f[\/latex]?<\/li>\n\t
                                                                                                              2. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.<\/li>\n\t
                                                                                                              3. If the graph of a polynomial just touches the [latex]x[\/latex]-axis and then changes direction, what can we conclude about the factored form of the polynomial?<\/li>\n<\/ol>\n

                                                                                                                Algebraic<\/h4>\n

                                                                                                                For the following exercises, find the [latex]x[\/latex]- or [latex]t[\/latex]-intercepts of the polynomial functions.<\/strong><\/p>\n

                                                                                                                  \n\t
                                                                                                                1. [latex]C(t)=3(t+2)(t\u22123)(t+5)[\/latex]<\/li>\n\t
                                                                                                                2. [latex]C(t)=2t(t\u22123)(t+1)^2[\/latex]<\/li>\n\t
                                                                                                                3. [latex]C(t)=4t^4+12t^3\u221240t^2[\/latex]<\/li>\n\t
                                                                                                                4. [latex]f(x)=x^3+x^2\u221220x[\/latex]<\/li>\n<\/ol>\n

                                                                                                                  For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.<\/strong><\/p>\n

                                                                                                                    \n\t
                                                                                                                  1. [latex]f(x)=x^3\u22129x[\/latex], between [latex]x=2[\/latex] and [latex]x=4[\/latex].<\/li>\n\t
                                                                                                                  2. [latex]f(x)=\u2212x^4+4[\/latex], between [latex]x=1[\/latex] and [latex]x=3[\/latex].<\/li>\n\t
                                                                                                                  3. [latex]f(x)=x^3\u2212100x+2[\/latex], between [latex]x=0.01[\/latex] and [latex]x=0.1[\/latex].<\/li>\n<\/ol>\n

                                                                                                                    For the following exercises, find the zeros and give the multiplicity of each.<\/strong><\/p>\n

                                                                                                                      \n\t
                                                                                                                    1. [latex]f(x)=x^2(2x+3)^5(x\u22124)^2[\/latex]<\/li>\n\t
                                                                                                                    2. [latex]f(x)=x^2(x^2+4x+4)[\/latex]<\/li>\n\t
                                                                                                                    3. [latex]f(x)=(3x+2)^5(x^2\u221210x+25)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                      Graphical<\/h4>\n

                                                                                                                      For the following exercises, graph the polynomial functions. Note [latex]x[\/latex]- and [latex]y[\/latex]- intercepts, multiplicity, and end behavior.<\/strong><\/p>\n

                                                                                                                        \n\t
                                                                                                                      1. [latex]g(x)=(x+4)(x\u22121)^2[\/latex]<\/li>\n\t
                                                                                                                      2. [latex]k(x)=(x\u22123)^3(x\u22122)^2[\/latex]<\/li>\n\t
                                                                                                                      3. [latex]n(x)=\u22123x(x+2)(x\u22124)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                        For the following exercises, use the graphs to write the formula for a polynomial function of least degree.<\/strong><\/p>\n

                                                                                                                          \n\t

                                                                                                                        1. \n\"Graph<\/li>\n\t

                                                                                                                        2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                          For the following exercises, use the graph to identify zeros and multiplicity.<\/strong><\/p>\n

                                                                                                                            \n\t

                                                                                                                          1. \n\"Graph<\/li>\n\t

                                                                                                                          2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                            For the following exercises, use the given information about the polynomial graph to write the equation.<\/strong><\/p>\n

                                                                                                                              \n\t
                                                                                                                            1. Degree [latex]3[\/latex]. Zeros at [latex]x=\u20132, x=1[\/latex], and [latex]x=3[\/latex]. [latex]y[\/latex]-intercept at [latex](0,\u20134)[\/latex].<\/li>\n\t
                                                                                                                            2. Degree [latex]5[\/latex]. Roots of multiplicity [latex]2[\/latex] at [latex]x=3[\/latex] and [latex]x=1[\/latex], and a root of multiplicity [latex]1[\/latex] at [latex]x=\u20133[\/latex]. [latex]y[\/latex]-intercept at [latex](0,9)[\/latex].<\/li>\n\t
                                                                                                                            3. Degree [latex]5[\/latex]. Double zero at [latex]x=1[\/latex], and triple zero at [latex]x=3[\/latex]. Passes through the point [latex](2,15)[\/latex].<\/li>\n<\/ol>\n

                                                                                                                              Technology<\/h4>\n

                                                                                                                              For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.<\/strong><\/p>\n

                                                                                                                                \n\t
                                                                                                                              1. [latex]f(x)=x^3\u2212x\u22121[\/latex]<\/li>\n\t
                                                                                                                              2. [latex]f(x)=x^4+x[\/latex]<\/li>\n\t
                                                                                                                              3. [latex]f(x)=x^4\u2212x^3+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                Dividing Polynomials<\/h2>\n

                                                                                                                                Algebraic<\/h4>\n

                                                                                                                                For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/strong><\/p>\n

                                                                                                                                  \n\t
                                                                                                                                1. [latex](x^2+5x\u22121)\u00f7(x\u22121)[\/latex]<\/li>\n\t
                                                                                                                                2. [latex](3x^2+23x+14)\u00f7(x+7)[\/latex]<\/li>\n\t
                                                                                                                                3. [latex](6x^2\u221225x\u221225)\u00f7(6x+5)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                  For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/strong><\/p>\n

                                                                                                                                    \n\t
                                                                                                                                  1. [latex](2x^3\u22126x^2\u22127x+6)\u00f7(x\u22124)[\/latex]<\/li>\n\t
                                                                                                                                  2. [latex](4x^3\u221212x^2\u22125x\u22121)\u00f7(2x+1)[\/latex]<\/li>\n\t
                                                                                                                                  3. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\n\t
                                                                                                                                  4. [latex](2x^3+7x^2\u221213x\u22123)\u00f7(2x\u22123)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                    For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/strong><\/p>\n

                                                                                                                                      \n\t
                                                                                                                                    1. [latex]x\u22122[\/latex], [latex]3x^4\u22126x^3\u22125x+10[\/latex]<\/li>\n\t
                                                                                                                                    2. [latex]x\u22122[\/latex], [latex]4x^4\u221215x^2\u22124[\/latex]<\/li>\n\t
                                                                                                                                    3. [latex]x+\\frac{1}{3}[\/latex], [latex]3x^4+x^3\u22123x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                      Graphical<\/h4>\n

                                                                                                                                      For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/strong><\/p>\n

                                                                                                                                        \n\t
                                                                                                                                      1. Factor is [latex](x^2+2x+4)[\/latex]
                                                                                                                                        \n\"Graph<\/li>\n\t
                                                                                                                                      2. Factor is [latex]x^2+x+1[\/latex]
                                                                                                                                        \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                        For the following exercises, use synthetic division to find the quotient and remainder.<\/strong><\/p>\n

                                                                                                                                          \n\t
                                                                                                                                        1. [latex]\\frac{4x^3\u221233}{x\u22122}[\/latex]<\/li>\n\t
                                                                                                                                        2. [latex]\\frac{3x^3+2x\u22125}{x\u22121}[\/latex]<\/li>\n\t
                                                                                                                                        3. [latex]\\frac{x^4\u221222}{x+2}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                          Zeros of Polynomial Functions<\/h2>\n

                                                                                                                                          Verbal<\/h4>\n
                                                                                                                                            \n\t
                                                                                                                                          1. Describe a use for the Remainder Theorem.<\/li>\n\t
                                                                                                                                          2. What is the difference between rational and real zeros?<\/li>\n\t
                                                                                                                                          3. If synthetic division reveals a zero, why should we try that value again as a possible solution?<\/li>\n<\/ol>\n

                                                                                                                                            Algebraic<\/h4>\n

                                                                                                                                            For the following exercises, use the Remainder Theorem to find the remainder.<\/strong><\/p>\n

                                                                                                                                              \n\t
                                                                                                                                            1. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\n\t
                                                                                                                                            2. [latex](\u22123x^2+6x+24)\u00f7(x\u22124)[\/latex]<\/li>\n\t
                                                                                                                                            3. [latex](x^4\u22121)\u00f7(x\u22124)[\/latex]<\/li>\n\t
                                                                                                                                            4. [latex](4x^3+5x^2\u22122x+7)\u00f7(x+2)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                              For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.<\/strong><\/p>\n

                                                                                                                                                \n\t
                                                                                                                                              1. [latex]f(x)=2x^3+x^2\u22125x+2; x+2[\/latex]<\/li>\n\t
                                                                                                                                              2. [latex]f(x)=2x^3+3x^2+x+6; x+2[\/latex]<\/li>\n\t
                                                                                                                                              3. [latex]x^3+3x^2+4x+12; x+3[\/latex]<\/li>\n\t
                                                                                                                                              4. [latex]2x^3+5x^2\u221212x\u221230; 2x+5[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.<\/strong><\/p>\n

                                                                                                                                                  \n\t
                                                                                                                                                1. [latex]2x^3+7x^2\u221210x\u221224=0[\/latex]<\/li>\n\t
                                                                                                                                                2. [latex]x^3+5x^2\u221216x\u221280=0[\/latex]<\/li>\n\t
                                                                                                                                                3. [latex]2x^3\u22123x^2\u221232x\u221215=0[\/latex]<\/li>\n\t
                                                                                                                                                4. [latex]2x^3\u22123x^2\u2212x+1=0[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                  Numeric<\/h4>\n

                                                                                                                                                  For the following exercises, list all possible rational zeros for the functions.<\/strong><\/p>\n

                                                                                                                                                    \n\t
                                                                                                                                                  1. [latex]f(x)=2x^3+3x^2\u22128x+5[\/latex]<\/li>\n\t
                                                                                                                                                  2. [latex]f(x)=6x^4\u221210x^2+13x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                    For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.<\/strong><\/p>\n

                                                                                                                                                      \n\t
                                                                                                                                                    1. [latex]f(x)=6x^3\u22127x^2+1[\/latex]<\/li>\n\t
                                                                                                                                                    2. [latex]f(x)=8x^3\u22126x^2\u221223x+6[\/latex]<\/li>\n<\/ol>\n","rendered":"

                                                                                                                                                      Functions and Function Notation<\/h2>\n

                                                                                                                                                      Verbal<\/h4>\n
                                                                                                                                                        \n
                                                                                                                                                      1. What is the difference between a relation and a function?<\/li>\n
                                                                                                                                                      2. Why does the vertical line test tell us whether the graph of a relation represents a function?<\/li>\n
                                                                                                                                                      3. Why does the horizontal line test tell us whether the graph of a function is one-to-one?<\/li>\n<\/ol>\n

                                                                                                                                                        Algebraic<\/h4>\n

                                                                                                                                                        For the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].<\/strong><\/p>\n

                                                                                                                                                          \n
                                                                                                                                                        1. [latex]y=x^2[\/latex]<\/li>\n
                                                                                                                                                        2. [latex]3x^2+y=14[\/latex]<\/li>\n
                                                                                                                                                        3. [latex]y=\u22122x^2+40x[\/latex]<\/li>\n
                                                                                                                                                        4. [latex]x=\\frac{3y+5}{7y\u22121}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                          For the following exercises, evaluate [latex]f(\u22123),f(2),f(\u2212a),\u2212f(a),f(a+h)[\/latex].<\/strong><\/p>\n

                                                                                                                                                            \n
                                                                                                                                                          1. [latex]f(x)=2x\u22125[\/latex]<\/li>\n
                                                                                                                                                          2. [latex]f(x)=\\sqrt{2\u2212x}+5[\/latex]<\/li>\n
                                                                                                                                                          3. [latex]f(x)=|x\u22121|\u2212|x+1|[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                            Graphical<\/h4>\n

                                                                                                                                                            For the following exercises, use the vertical line test to determine which graphs show relations that are functions.<\/strong><\/p>\n

                                                                                                                                                              \n
                                                                                                                                                            1. \n\"Graph<\/li>\n
                                                                                                                                                            2. \n\"Graph<\/li>\n
                                                                                                                                                            3. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                              For the following exercises, determine if the given graph is a one-to-one function.<\/strong><\/p>\n

                                                                                                                                                                \n
                                                                                                                                                              1. \n\"Graph<\/li>\n
                                                                                                                                                              2. \n\"Graph<\/li>\n
                                                                                                                                                              3. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                Numeric<\/h4>\n

                                                                                                                                                                For the following exercises, determine whether the relation represents a function.<\/strong><\/p>\n

                                                                                                                                                                  \n
                                                                                                                                                                1. [latex]{(3,4),(4,5),(5,6)}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                  For the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].<\/strong><\/p>\n

                                                                                                                                                                    \n
                                                                                                                                                                  1. \n\n\n\n\n
                                                                                                                                                                    [latex]x[\/latex]<\/strong><\/td>\n5<\/td>\n10<\/td>\n15<\/td>\n<\/tr>\n
                                                                                                                                                                    [latex]y[\/latex]<\/strong><\/td>\n3<\/td>\n8<\/td>\n14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
                                                                                                                                                                  2. \n\n\n\n\n
                                                                                                                                                                    [latex]x[\/latex]<\/strong><\/td>\n5<\/td>\n10<\/td>\n10<\/td>\n<\/tr>\n
                                                                                                                                                                    [latex]y[\/latex]<\/strong><\/td>\n3<\/td>\n8<\/td>\n14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                                                                                                                    For the following exercise, use the function [latex]f[\/latex] represented in the table below.<\/strong><\/p>\n

                                                                                                                                                                      \n
                                                                                                                                                                    1. Solve [latex]f(x)=1[\/latex].
                                                                                                                                                                      \n\n\n\n\n
                                                                                                                                                                      [latex]x[\/latex]<\/strong><\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n<\/tr>\n
                                                                                                                                                                      [latex]f(x)[\/latex]<\/strong><\/td>\n74<\/td>\n28<\/td>\n1<\/td>\n53<\/td>\n56<\/td>\n3<\/td>\n36<\/td>\n45<\/td>\n14<\/td>\n47<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                                                                                                                      For the following exercises, evaluate the function [latex]f[\/latex] at the values [latex]f(\u22122),f(\u22121),f(0),f(1),[\/latex] and [latex]f(2)[\/latex].<\/strong><\/p>\n

                                                                                                                                                                        \n
                                                                                                                                                                      1. [latex]f(x)=f(x)=8\u22123x[\/latex]<\/li>\n
                                                                                                                                                                      2. [latex]f(x)=3+\\sqrt{x+3}[\/latex]<\/li>\n
                                                                                                                                                                      3. [latex]f(x)=3^x[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                        For the following exercise, evaluate the expression, given functions [latex]f[\/latex],[latex]g[\/latex], and [latex]h[\/latex]: <\/strong><\/p>\n

                                                                                                                                                                        [latex]f(x)=3x\u22122[\/latex]<\/strong><\/p>\n

                                                                                                                                                                        [latex]g(x)=5\u2212x^2[\/latex]<\/strong><\/p>\n

                                                                                                                                                                        [latex]h(x)=\u22122x^2+3x\u22121[\/latex] <\/strong><\/p>\n

                                                                                                                                                                          \n
                                                                                                                                                                        1. [latex]f(\\frac{7}{3})\u2212h(\u22122)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                          For the following exercise, graph [latex]y=x^2[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                                                                                                                                                                            \n
                                                                                                                                                                          1. [latex][\u221210,10][\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                            For the following exercises, graph [latex]y=x^3[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                                                                                                                                                                              \n
                                                                                                                                                                            1. [latex][\u22120.1,0.1][\/latex]<\/li>\n
                                                                                                                                                                            2. [latex][\u2212100,100][\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                              For the following exercise, graph [latex]y=\\sqrt{x}[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                                                                                                                                                                                \n
                                                                                                                                                                              1. [latex][0,100][\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                For the following exercises, graph [latex]y=\\sqrt[3]{x}[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/strong><\/p>\n

                                                                                                                                                                                  \n
                                                                                                                                                                                1. [latex][\u22120.001, 0.001][\/latex]<\/li>\n
                                                                                                                                                                                2. [latex][\u22121,000,000, 1,000,000][\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                  Real-World Applications<\/strong><\/p>\n

                                                                                                                                                                                    \n
                                                                                                                                                                                  1. The number of cubic yards of dirt, [latex]D[\/latex], needed to cover a garden with area [latex]a[\/latex] square feet is given by [latex]D=g(a)[\/latex].\n
                                                                                                                                                                                      \n
                                                                                                                                                                                    1. A garden with area [latex]5000 ft^2[\/latex] requires [latex]50 yd^3[\/latex] of dirt. Express this information in terms of the function [latex]g[\/latex].<\/li>\n
                                                                                                                                                                                    2. Explain the meaning of the statement [latex]g(100)=1[\/latex].<\/li>\n<\/ol>\n<\/li>\n
                                                                                                                                                                                    3. Let [latex]h(t)[\/latex] be the height above ground, in feet, of a rocket [latex]t[\/latex] seconds after launching. Explain the meaning of each statement:\n
                                                                                                                                                                                        \n
                                                                                                                                                                                      1. [latex]h(1)=200[\/latex]<\/li>\n
                                                                                                                                                                                      2. [latex]h(2)=350[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                                                                                                                                        Linear Functions<\/h2>\n

                                                                                                                                                                                        Verbal<\/h4>\n
                                                                                                                                                                                          \n
                                                                                                                                                                                        1. Terry is skiing down a steep hill. Terry’s elevation, [latex]E(t)[\/latex], in feet after t seconds is given by [latex]E(t)=3000\u221270t[\/latex]. Write a complete sentence describing Terry\u2019s starting elevation and how it is changing over time.<\/li>\n
                                                                                                                                                                                        2. A boat is [latex]100[\/latex] miles away from the marina, sailing directly toward it at [latex]10[\/latex] miles per hour. Write an equation for the distance of the boat from the marina after [latex]t[\/latex]hours.<\/li>\n<\/ol>\n

                                                                                                                                                                                          Algebraic<\/h4>\n

                                                                                                                                                                                          For the following exercises, determine whether the equation of the curve can be written as a linear function.<\/strong><\/p>\n

                                                                                                                                                                                            \n
                                                                                                                                                                                          1. [latex]y=3x-5[\/latex]<\/li>\n
                                                                                                                                                                                          2. [latex]3x+5y=15[\/latex]<\/li>\n
                                                                                                                                                                                          3. [latex]3x+5y^2=15[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                            For the following exercises, determine whether each function is increasing or decreasing.<\/strong><\/p>\n

                                                                                                                                                                                              \n
                                                                                                                                                                                            1. [latex]g(x)=5x+6[\/latex]<\/li>\n
                                                                                                                                                                                            2. [latex]b(x)=8\u22123x[\/latex]<\/li>\n
                                                                                                                                                                                            3. [latex]k(x)=\u22124x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                              For the following exercises, find the slope of the line that passes through the two given points.<\/strong><\/p>\n

                                                                                                                                                                                                \n
                                                                                                                                                                                              1. [latex](1,5)[\/latex] and [latex](4,11)[\/latex]<\/li>\n
                                                                                                                                                                                              2. [latex](8,\u20132)[\/latex] and [latex](4,6)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.<\/strong><\/p>\n

                                                                                                                                                                                                  \n
                                                                                                                                                                                                1. [latex]f(\u22125)=\u22124[\/latex] and [latex]f(5)=2[\/latex]<\/li>\n
                                                                                                                                                                                                2. Passes through [latex](2,4)[\/latex] and [latex](4,10)[\/latex]<\/li>\n
                                                                                                                                                                                                3. Passes through [latex](-1,4)[\/latex] and [latex](5,2)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                  For the following exercises, find the x<\/em>– and y-<\/em>intercepts of each equation.<\/strong><\/p>\n

                                                                                                                                                                                                    \n
                                                                                                                                                                                                  1. [latex]f(x)=\u2212x+2[\/latex]<\/li>\n
                                                                                                                                                                                                  2. [latex]h(x)=3x\u22125[\/latex]<\/li>\n
                                                                                                                                                                                                  3. [latex]\u22122x+5y=20[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                    For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. <\/strong><\/p>\n

                                                                                                                                                                                                      \n
                                                                                                                                                                                                    1. Line 1: Passes through [latex](0,6)[\/latex] and [latex](3,\u221224)[\/latex]
                                                                                                                                                                                                      \nLine 2: Passes through [latex](\u22121,19)[\/latex] and [latex](8,\u221271)[\/latex]<\/li>\n
                                                                                                                                                                                                    2. Line 1: Passes through [latex](2,3)[\/latex] and [latex](4,\u22121)[\/latex]
                                                                                                                                                                                                      \nLine 2: Passes through [latex](6,3)[\/latex] and [latex](8,5)[\/latex]<\/li>\n
                                                                                                                                                                                                    3. Line 1: Passes through [latex](2,5)[\/latex] and [latex](5,\u22121)[\/latex]
                                                                                                                                                                                                      \nLine 2: Passes through [latex](\u22123,7)[\/latex] and [latex](3,\u22125)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                      For the following exercises, write an equation for the line described.<\/strong><\/p>\n

                                                                                                                                                                                                        \n
                                                                                                                                                                                                      1. Write an equation for a line parallel to [latex]g(x)=3x\u22121[\/latex] and passing through the point [latex](4,9)[\/latex].<\/li>\n
                                                                                                                                                                                                      2. Write an equation for a line perpendicular to [latex]p(t)=3t+4[\/latex] and passing through the point [latex](3,1)[\/latex].<\/li>\n<\/ol>\n

                                                                                                                                                                                                        Graphical<\/h4>\n

                                                                                                                                                                                                        For the following exercise, find the slope of the line graphed.<\/strong><\/p>\n

                                                                                                                                                                                                          \n
                                                                                                                                                                                                        1. \n\"This<\/li>\n<\/ol>\n

                                                                                                                                                                                                          For the following exercises, write an equation for the line graphed.<\/strong><\/p>\n

                                                                                                                                                                                                            \n
                                                                                                                                                                                                          1. \n\"Graph<\/li>\n
                                                                                                                                                                                                          2. \n\"Graph<\/li>\n
                                                                                                                                                                                                          3. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                            For the following exercises, match the given linear equation with its graph in the figure below.<\/strong><\/p>\n

                                                                                                                                                                                                            \"Graph<\/div>\n
                                                                                                                                                                                                              \n
                                                                                                                                                                                                            1. [latex]f(x)=\u22123x\u22121[\/latex]<\/li>\n
                                                                                                                                                                                                            2. [latex]f(x)=2[\/latex]<\/li>\n
                                                                                                                                                                                                            3. [latex]f(x)=3x+2[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                              For the following exercises, sketch a line with the given features.<\/strong><\/p>\n

                                                                                                                                                                                                                \n
                                                                                                                                                                                                              1. An x-intercept [latex](\u20132,0)[\/latex] and y-intercept of [latex](0,4)[\/latex]<\/li>\n
                                                                                                                                                                                                              2. A y-intercept of [latex](0,3)[\/latex] and slope [latex]\\frac{2}{5}[\/latex]<\/li>\n
                                                                                                                                                                                                              3. Passing through the points [latex](\u20133,\u20134)[\/latex] and [latex](3,0)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                For the following exercises, sketch the graph of each equation.<\/strong><\/p>\n

                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                1. [latex]f(x)=\u22123x+2[\/latex]<\/li>\n
                                                                                                                                                                                                                2. [latex]f(x)=\\frac{2}{3}x\u22123[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                  For the following exercises, write the equation of the line shown in the graph.<\/strong><\/p>\n

                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                  1. \n\"The<\/li>\n
                                                                                                                                                                                                                  2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                                    Numeric<\/h4>\n

                                                                                                                                                                                                                    For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.<\/strong><\/p>\n

                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                    1. \n\n\n\n\n
                                                                                                                                                                                                                      [latex]x[\/latex]<\/strong><\/td>\n[latex]0[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n[latex]15[\/latex]<\/td>\n<\/tr>\n
                                                                                                                                                                                                                      [latex]g(x)[\/latex]<\/strong><\/td>\n[latex]5[\/latex]<\/td>\n[latex]-10[\/latex]<\/td>\n[latex]-25[\/latex]<\/td>\n[latex]-40[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
                                                                                                                                                                                                                    2. \n\n\n\n\n
                                                                                                                                                                                                                      [latex]x[\/latex]<\/strong><\/td>\n[latex]0[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n[latex]15[\/latex]<\/td>\n<\/tr>\n
                                                                                                                                                                                                                      [latex]f(x)[\/latex]<\/td>\n[latex]-5[\/latex]<\/td>\n[latex]20[\/latex]<\/td>\n[latex]45[\/latex]<\/td>\n[latex]70[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                                                                                                                                                                      Technology<\/h4>\n

                                                                                                                                                                                                                      For the following exercises, use a calculator or graphing technology to complete the task.<\/strong><\/p>\n

                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                      1. If [latex]f[\/latex] is a linear function, [latex]f(0.1)=11.5[\/latex], and [latex]f(0.4)=\u20135.9[\/latex], find an equation for the function.<\/li>\n
                                                                                                                                                                                                                      2. Graph the function [latex]f[\/latex] on a domain of [latex][\u201310,10]: f(x)=2,500x+4,000[\/latex].<\/li>\n
                                                                                                                                                                                                                      3. The table shows the input, [latex]p[\/latex], and output, [latex]q[\/latex], for a linear function [latex]q[\/latex].\n
                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                        1. Fill in the missing values of the table.<\/li>\n
                                                                                                                                                                                                                        2. Write the linear function [latex]q[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n\n
                                                                                                                                                                                                                          [latex]p[\/latex]<\/strong><\/td>\n[latex]0.5[\/latex]<\/td>\n[latex]0.8[\/latex]<\/td>\n[latex]12[\/latex]<\/td>\n[latex]b[\/latex]<\/td>\n<\/tr>\n
                                                                                                                                                                                                                          [latex]q[\/latex]<\/strong><\/td>\n[latex]400[\/latex]<\/td>\n[latex]700[\/latex]<\/td>\n[latex]a[\/latex]<\/td>\n[latex]1,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                          1. Graph the linear function [latex]f[\/latex] on a domain of [latex][\u22120.1,0.1][\/latex] for the function whose slope is [latex]75[\/latex] and y-intercept is [latex]\u221222.5[\/latex]. Label the points for the input values of [latex]\u22120.1[\/latex] and [latex]0.1[\/latex].<\/li>\n<\/ol>\n

                                                                                                                                                                                                                            Real-World Applications<\/h4>\n
                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                            1. A gym membership with two personal training sessions costs [latex]$125[\/latex], while gym membership with five personal training sessions costs [latex]$260[\/latex]. What is cost per session?<\/li>\n
                                                                                                                                                                                                                            2. A phone company charges for service according to the formula: [latex]C(n)=24+0.1n[\/latex], where [latex]n[\/latex] is the number of minutes talked, and [latex]C(n)[\/latex] is the monthly charge, in dollars. Find and interpret the rate of change and initial value.<\/li>\n
                                                                                                                                                                                                                            3. A city\u2019s population in the year 1960 was [latex]287,500[\/latex]. In 1989 the population was [latex]275,900[\/latex]. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.<\/li>\n
                                                                                                                                                                                                                            4. Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: [latex]I(x)=1054x+23,286[\/latex], where [latex]x[\/latex] is the number of years after 1990. Which of the following interprets the slope in the context of the problem?\n
                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                              1. As of 1990, average annual income was [latex]$23,286[\/latex].<\/li>\n
                                                                                                                                                                                                                              2. In the ten-year period from 1990\u20131999, average annual income increased by a total of [latex]$1,054[\/latex].<\/li>\n
                                                                                                                                                                                                                              3. Each year in the decade of the 1990s, average annual income increased by [latex]$1,054[\/latex].<\/li>\n
                                                                                                                                                                                                                              4. Average annual income rose to a level of [latex]$23,286[\/latex] by the end of 1999.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                Quadratic Functions<\/h2>\n

                                                                                                                                                                                                                                Verbal<\/h4>\n
                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                1. Explain the advantage of writing a quadratic function in standard form.<\/li>\n
                                                                                                                                                                                                                                2. Explain why the condition of [latex]a\u22600[\/latex] is imposed in the definition of the quadratic function.<\/li>\n
                                                                                                                                                                                                                                3. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                  Algebraic<\/h4>\n

                                                                                                                                                                                                                                  For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/strong><\/p>\n

                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                  1. [latex]g(x)=x^2+2x\u22123[\/latex]<\/li>\n
                                                                                                                                                                                                                                  2. [latex]f(x)=x^2+5x\u22122[\/latex]<\/li>\n
                                                                                                                                                                                                                                  3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                    For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/strong><\/p>\n

                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                    1. [latex]f(x)=2x^2\u221210x+4[\/latex]<\/li>\n
                                                                                                                                                                                                                                    2. [latex]f(x)=4x^2+x\u22121[\/latex]<\/li>\n
                                                                                                                                                                                                                                    3. [latex]f(x)=\\frac{1}{2}x^2+3x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                      For the following exercises, determine the domain and range of the quadratic function.<\/strong><\/p>\n

                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                      1. [latex]f(x)=(x\u22123)^2+2[\/latex]<\/li>\n
                                                                                                                                                                                                                                      2. [latex]f(x)=x^2+6x+4[\/latex]<\/li>\n
                                                                                                                                                                                                                                      3. [latex]k(x)=3x^2\u22126x\u22129[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                        For the following exercises, use the vertex [latex](h,k)[\/latex] and a point on the graph [latex](x,y)[\/latex] to find the general form of the equation of the quadratic function.<\/strong><\/p>\n

                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                        1. [latex](h,k)=(\u22122,\u22121),(x,y)=(\u22124,3)[\/latex]<\/li>\n
                                                                                                                                                                                                                                        2. [latex](h,k)=(2,3),(x,y)=(5,12)[\/latex]<\/li>\n
                                                                                                                                                                                                                                        3. [latex](h,k)=(3,2),(x,y)=(10,1)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                          Graphical<\/h4>\n

                                                                                                                                                                                                                                          For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/strong><\/p>\n

                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                          1. [latex]f(x)=x^2\u22126x\u22121[\/latex]<\/li>\n
                                                                                                                                                                                                                                          2. [latex]f(x)=x^2\u22127x+3[\/latex]<\/li>\n
                                                                                                                                                                                                                                          3. [latex]f(x)=4x^2\u221212x\u22123[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                            For the following exercises, write the equation for the graphed quadratic function.<\/strong><\/p>\n

                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                            1. \n\"Graph<\/li>\n
                                                                                                                                                                                                                                            2. \n\"Graph<\/p>\n<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                              Numeric<\/h4>\n

                                                                                                                                                                                                                                              For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.<\/strong><\/p>\n

                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                              1. \n\n\n\n\n
                                                                                                                                                                                                                                                [latex]x[\/latex]<\/strong><\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                [latex]y[\/latex]<\/strong><\/td>\n[latex]1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
                                                                                                                                                                                                                                              2. \n\n\n\n\n
                                                                                                                                                                                                                                                [latex]x[\/latex]<\/strong><\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                [latex]y[\/latex]<\/strong><\/td>\n[latex]-8[\/latex]<\/td>\n[latex]-3[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                Power Functions and Polynomial Functions<\/h2>\n

                                                                                                                                                                                                                                                Verbal<\/h4>\n
                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                1. Explain the difference between the coefficient of a power function and its degree.<\/li>\n
                                                                                                                                                                                                                                                2. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/li>\n
                                                                                                                                                                                                                                                3. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As [latex]x\u2192\u2212\u221e[\/latex], [latex]f(x)\u2192\u2212\u221e[\/latex] and as [latex]x\u2192\u221e[\/latex], [latex]f(x)\u2192\u2212\u221e[\/latex].<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                  Algebraic<\/h4>\n

                                                                                                                                                                                                                                                  For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/strong><\/p>\n

                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                  1. [latex]f(x)=(x^2)^3[\/latex]<\/li>\n
                                                                                                                                                                                                                                                  2. [latex]f(x)=\\frac{x^2}{x^2\u22121}[\/latex]<\/li>\n
                                                                                                                                                                                                                                                  3. [latex]f(x)=3^{x+1}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                    For the following exercises, find the degree and leading coefficient for the given polynomial.<\/strong><\/p>\n

                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                    1. [latex]7\u22122x^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                    2. [latex]x(4\u2212x^2)(2x+1)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                      For the following exercises, determine the end behavior of the functions.<\/strong><\/p>\n

                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                      1. [latex]f(x)=x^4[\/latex]<\/li>\n
                                                                                                                                                                                                                                                      2. [latex]f(x)=\u2212x^4[\/latex]<\/li>\n
                                                                                                                                                                                                                                                      3. [latex]f(x)=\u22122x^4\u22123x^2+x\u22121[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                        For the following exercises, find the intercepts of the functions.<\/strong><\/p>\n

                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                        1. [latex]f(t)=2(t\u22121)(t+2)(t\u22123)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                        2. [latex]f(x)=x^4\u221216[\/latex]<\/li>\n
                                                                                                                                                                                                                                                        3. [latex]f(x)=x(x^2\u22122x\u22128)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                          Graphical<\/h4>\n

                                                                                                                                                                                                                                                          For the following exercises, determine the least possible degree of the polynomial function shown.<\/strong><\/p>\n

                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                          1. \n\"Graph<\/li>\n
                                                                                                                                                                                                                                                          2. \n\"Graph<\/li>\n
                                                                                                                                                                                                                                                          3. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                            For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/strong><\/p>\n

                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                            1. \n\"Graph<\/li>\n
                                                                                                                                                                                                                                                            2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                              Numeric<\/h4>\n

                                                                                                                                                                                                                                                              For the following exercises, make a table to confirm the end behavior of the function.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                              1. [latex]f(x)=x^4\u22125x^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                              2. [latex]f(x)=(x\u22121)(x\u22122)(3\u2212x)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                Technology<\/h4>\n

                                                                                                                                                                                                                                                                For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                1. [latex]f(x)=x^3(x\u22122)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                2. [latex]f(x)=x(14\u22122x)(10\u22122x)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                3. [latex]f(x)=x^3\u221216x[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                  Graphs of Polynomial Functions<\/h2>\n

                                                                                                                                                                                                                                                                  Verbal<\/h4>\n
                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                  1. What is the difference between an [latex]x[\/latex]– intercept and a zero of a polynomial function [latex]f[\/latex]?<\/li>\n
                                                                                                                                                                                                                                                                  2. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.<\/li>\n
                                                                                                                                                                                                                                                                  3. If the graph of a polynomial just touches the [latex]x[\/latex]-axis and then changes direction, what can we conclude about the factored form of the polynomial?<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                    Algebraic<\/h4>\n

                                                                                                                                                                                                                                                                    For the following exercises, find the [latex]x[\/latex]– or [latex]t[\/latex]-intercepts of the polynomial functions.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                    1. [latex]C(t)=3(t+2)(t\u22123)(t+5)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                    2. [latex]C(t)=2t(t\u22123)(t+1)^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                    3. [latex]C(t)=4t^4+12t^3\u221240t^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                    4. [latex]f(x)=x^3+x^2\u221220x[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                      For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                      1. [latex]f(x)=x^3\u22129x[\/latex], between [latex]x=2[\/latex] and [latex]x=4[\/latex].<\/li>\n
                                                                                                                                                                                                                                                                      2. [latex]f(x)=\u2212x^4+4[\/latex], between [latex]x=1[\/latex] and [latex]x=3[\/latex].<\/li>\n
                                                                                                                                                                                                                                                                      3. [latex]f(x)=x^3\u2212100x+2[\/latex], between [latex]x=0.01[\/latex] and [latex]x=0.1[\/latex].<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                        For the following exercises, find the zeros and give the multiplicity of each.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                        1. [latex]f(x)=x^2(2x+3)^5(x\u22124)^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                        2. [latex]f(x)=x^2(x^2+4x+4)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                        3. [latex]f(x)=(3x+2)^5(x^2\u221210x+25)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                          Graphical<\/h4>\n

                                                                                                                                                                                                                                                                          For the following exercises, graph the polynomial functions. Note [latex]x[\/latex]– and [latex]y[\/latex]– intercepts, multiplicity, and end behavior.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                          1. [latex]g(x)=(x+4)(x\u22121)^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                          2. [latex]k(x)=(x\u22123)^3(x\u22122)^2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                          3. [latex]n(x)=\u22123x(x+2)(x\u22124)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                            For the following exercises, use the graphs to write the formula for a polynomial function of least degree.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                            1. \n\"Graph<\/li>\n
                                                                                                                                                                                                                                                                            2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                              For the following exercises, use the graph to identify zeros and multiplicity.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                              1. \n\"Graph<\/li>\n
                                                                                                                                                                                                                                                                              2. \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                For the following exercises, use the given information about the polynomial graph to write the equation.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                1. Degree [latex]3[\/latex]. Zeros at [latex]x=\u20132, x=1[\/latex], and [latex]x=3[\/latex]. [latex]y[\/latex]-intercept at [latex](0,\u20134)[\/latex].<\/li>\n
                                                                                                                                                                                                                                                                                2. Degree [latex]5[\/latex]. Roots of multiplicity [latex]2[\/latex] at [latex]x=3[\/latex] and [latex]x=1[\/latex], and a root of multiplicity [latex]1[\/latex] at [latex]x=\u20133[\/latex]. [latex]y[\/latex]-intercept at [latex](0,9)[\/latex].<\/li>\n
                                                                                                                                                                                                                                                                                3. Degree [latex]5[\/latex]. Double zero at [latex]x=1[\/latex], and triple zero at [latex]x=3[\/latex]. Passes through the point [latex](2,15)[\/latex].<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                  Technology<\/h4>\n

                                                                                                                                                                                                                                                                                  For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                  1. [latex]f(x)=x^3\u2212x\u22121[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                  2. [latex]f(x)=x^4+x[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                  3. [latex]f(x)=x^4\u2212x^3+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                    Dividing Polynomials<\/h2>\n

                                                                                                                                                                                                                                                                                    Algebraic<\/h4>\n

                                                                                                                                                                                                                                                                                    For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                    1. [latex](x^2+5x\u22121)\u00f7(x\u22121)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                    2. [latex](3x^2+23x+14)\u00f7(x+7)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                    3. [latex](6x^2\u221225x\u221225)\u00f7(6x+5)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                      For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                      1. [latex](2x^3\u22126x^2\u22127x+6)\u00f7(x\u22124)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                      2. [latex](4x^3\u221212x^2\u22125x\u22121)\u00f7(2x+1)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                      3. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                      4. [latex](2x^3+7x^2\u221213x\u22123)\u00f7(2x\u22123)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                        For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                        1. [latex]x\u22122[\/latex], [latex]3x^4\u22126x^3\u22125x+10[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                        2. [latex]x\u22122[\/latex], [latex]4x^4\u221215x^2\u22124[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                        3. [latex]x+\\frac{1}{3}[\/latex], [latex]3x^4+x^3\u22123x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                          Graphical<\/h4>\n

                                                                                                                                                                                                                                                                                          For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                          1. Factor is [latex](x^2+2x+4)[\/latex]
                                                                                                                                                                                                                                                                                            \n\"Graph<\/li>\n
                                                                                                                                                                                                                                                                                          2. Factor is [latex]x^2+x+1[\/latex]
                                                                                                                                                                                                                                                                                            \n\"Graph<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                            For the following exercises, use synthetic division to find the quotient and remainder.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                            1. [latex]\\frac{4x^3\u221233}{x\u22122}[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                            2. [latex]\\frac{3x^3+2x\u22125}{x\u22121}[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                            3. [latex]\\frac{x^4\u221222}{x+2}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                              Zeros of Polynomial Functions<\/h2>\n

                                                                                                                                                                                                                                                                                              Verbal<\/h4>\n
                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                              1. Describe a use for the Remainder Theorem.<\/li>\n
                                                                                                                                                                                                                                                                                              2. What is the difference between rational and real zeros?<\/li>\n
                                                                                                                                                                                                                                                                                              3. If synthetic division reveals a zero, why should we try that value again as a possible solution?<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                                Algebraic<\/h4>\n

                                                                                                                                                                                                                                                                                                For the following exercises, use the Remainder Theorem to find the remainder.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                1. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                2. [latex](\u22123x^2+6x+24)\u00f7(x\u22124)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                3. [latex](x^4\u22121)\u00f7(x\u22124)[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                4. [latex](4x^3+5x^2\u22122x+7)\u00f7(x+2)[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                                  For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                                  1. [latex]f(x)=2x^3+x^2\u22125x+2; x+2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                  2. [latex]f(x)=2x^3+3x^2+x+6; x+2[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                  3. [latex]x^3+3x^2+4x+12; x+3[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                  4. [latex]2x^3+5x^2\u221212x\u221230; 2x+5[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                                    For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                    1. [latex]2x^3+7x^2\u221210x\u221224=0[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                    2. [latex]x^3+5x^2\u221216x\u221280=0[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                    3. [latex]2x^3\u22123x^2\u221232x\u221215=0[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                    4. [latex]2x^3\u22123x^2\u2212x+1=0[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                                      Numeric<\/h4>\n

                                                                                                                                                                                                                                                                                                      For the following exercises, list all possible rational zeros for the functions.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                      1. [latex]f(x)=2x^3+3x^2\u22128x+5[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                      2. [latex]f(x)=6x^4\u221210x^2+13x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                                        For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                        1. [latex]f(x)=6x^3\u22127x^2+1[\/latex]<\/li>\n
                                                                                                                                                                                                                                                                                                        2. [latex]f(x)=8x^3\u22126x^2\u221223x+6[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":6,"menu_order":48,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/239"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/239\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/239\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=239"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=239"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=239"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}