{"id":8420,"date":"2023-09-29T14:53:15","date_gmt":"2023-09-29T14:53:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8420"},"modified":"2023-11-26T23:51:45","modified_gmt":"2023-11-26T23:51:45","slug":"functions-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/functions-cheat-sheet\/","title":{"raw":"Functions: Cheat Sheet","rendered":"Functions: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<ul>\r\n\t<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\r\n\t<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right)[\/latex].<\/li>\r\n\t<li>In table form, a function can be represented by rows or columns that relate to input and output values.<\/li>\r\n\t<li>To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.<\/li>\r\n\t<li>To solve for a specific function value, we determine the input values that yield the specific output value.<\/li>\r\n\t<li>An algebraic form of a function can be written from an equation.<\/li>\r\n\t<li>Input and output values of a function can be identified from a table.<\/li>\r\n\t<li>Relating input values to output values on a graph is another way to evaluate a function.<\/li>\r\n\t<li>A function is one-to-one if each output value corresponds to only one input value.<\/li>\r\n\t<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\r\n\t<li>A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.<\/li>\r\n\t<li>Fundamental toolkit functions include linear, quadratic, polynomial, exponential, and logarithmic functions, among others. Each type of function has a distinct graph shape. For example, linear functions have straight-line graphs, quadratic functions have parabolic graphs, and exponential functions have curved graphs that increase or decrease rapidly.<\/li>\r\n\t<li>The ordered pairs given by a linear function represent points on a line.<\/li>\r\n\t<li>Linear functions can be represented in words, function notation, tabular form and graphical form.<\/li>\r\n\t<li>The rate of change of a linear function is also known as the slope.<\/li>\r\n\t<li>An equation in slope-intercept form of a line includes the slope and the initial value of the function.<\/li>\r\n\t<li>The initial value, or [latex]y[\/latex]-intercept, is the output value when the input of a linear function is zero. It is the [latex]y[\/latex]-value of the point where the line crosses the [latex]y[\/latex]-axis.<\/li>\r\n\t<li>An increasing linear function results in a graph that slants upward from left to right and has a positive slope.\u00a0A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line.<\/li>\r\n\t<li>Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.<\/li>\r\n\t<li>The slope of a linear function can be calculated by dividing the difference between [latex]y[\/latex]-values by the difference in corresponding [latex]x[\/latex]-values of any two points on the line.<\/li>\r\n\t<li>The slope and initial value can be determined given a graph or any two points on the line.<\/li>\r\n\t<li>Point-slope form is useful for finding the equation of a linear function when given the slope of a line and one point. It is also convenient for finding the equation of a linear function when given two points through which a line passes.<\/li>\r\n\t<li>The equation for a linear function can be written in slope-intercept form if the slope [latex]m[\/latex]\u00a0and initial value [latex]b[\/latex] are known.<\/li>\r\n\t<li>Linear functions may be graphed by plotting points or by using the <em>y<\/em>-intercept and slope.<\/li>\r\n\t<li>Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections.<\/li>\r\n\t<li>The <em>y<\/em>-intercept and slope of a line may be used to write the equation of a line. The <em>x<\/em>-intercept is the point at which the graph of a linear function crosses the <em>x<\/em>-axis.<\/li>\r\n\t<li>Horizontal lines are written in the form, [latex]f(x)=b[\/latex]. Vertical lines are written in the form, [latex]x=b[\/latex].<\/li>\r\n\t<li>Transformations involve shifting, stretching, or compressing the graph of a function in various ways. For linear functions, these transformations can include vertical shifts, and vertical stretches or compressions.\r\n\r\n<ul>\r\n\t<li>Vertical Shift: Changing the y-intercept [latex]b[\/latex] in [latex]f(x) = mx + b[\/latex] results in a vertical shift of the line. If [latex]b[\/latex] increases, the line shifts up; if [latex]b[\/latex] decreases, it shifts down.<\/li>\r\n\t<li>Vertical Stretch or Compression: Altering the slope [latex]m[\/latex] affects the steepness of the line. A larger absolute value of [latex]m[\/latex] indicates a steeper slope. A positive [latex]m[\/latex] indicates an upward slope, while a negative [latex]m[\/latex] indicates a downward slope.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>A polynomial function of degree two is called a quadratic function.<\/li>\r\n\t<li>The axis of symmetry is the vertical line passing through the vertex.<\/li>\r\n\t<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\r\n\t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\r\n\t<li>The domain of a quadratic function is all real numbers. The range varies with the function.\r\n\r\n<ul>\r\n\t<li>If the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex].<\/li>\r\n\t<li>If the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.\r\n\r\n<ul>\r\n\t<li>The parabola opens upward if [latex]a &gt; 0[\/latex] and downward if [latex]a &lt; 0[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Transformations of quadratic functions\r\n\r\n<ul>\r\n\t<li>Vertical Shift General Form: Changing the constant [latex]c[\/latex] in the equation results in a vertical shift of the parabola. If [latex]c[\/latex] increases, the parabola shifts upwards; if [latex]c[\/latex] decreases, it shifts downwards.<\/li>\r\n\t<li>Vertical Shift Standard (Vertex) Form: Changing [latex]k[\/latex] results in a vertical shift of the entire parabola. Increasing [latex]k[\/latex] shifts the parabola upwards, while decreasing [latex]k[\/latex] shifts it downwards.<\/li>\r\n\t<li>Horizontal Shift: This can be achieved by substituting [latex]x[\/latex] with [latex](x - h)[\/latex], resulting in a function of the form [latex]f(x) = a(x - h)^2 + k[\/latex]. The parabola shifts to the right if [latex]h[\/latex] is positive and to the left if [latex]h[\/latex] is negative.<\/li>\r\n\t<li>Stretch\/Compression: Changing the value of [latex]a[\/latex] causes a vertical stretch or compression. If the absolute value of [latex]a[\/latex] is greater than 1, the parabola becomes narrower (stretches), and if it's between 0 and 1, the parabola becomes wider (compresses).<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>A polynomial function is one whose equation contains only non-negative integer powers on the variable.<\/li>\r\n\t<li>The polynomial term containing the highest power on the variable is the leading term, and its degree is the number of the power. The leading coefficient of a polynomial is the coefficient of the leading term.<\/li>\r\n\t<li>The graph of a polynomial function describes a smooth, continuous curve.<\/li>\r\n\t<li>The domain of all polynomial functions is all real numbers.<\/li>\r\n\t<li>Even degree polynomial functions describe graphs whose ends both point up or both point down.<\/li>\r\n\t<li>Odd degree polynomial functions describe graphs whose ends points in opposite directions.<\/li>\r\n\t<li>The sign of the leading term will determine the direction of the ends of the graph:\r\n\r\n<ul>\r\n\t<li>even degree and positive coefficient: both ends point up<\/li>\r\n\t<li>even degree and negative coefficient: both ends point down<\/li>\r\n\t<li>odd degree and positive coefficient: the left-most end points down and the right-most end points up.<\/li>\r\n\t<li>odd degree and negative coefficient: the left-most end points up and the right-most end points down.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\r\n\t<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\r\n\t<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form [latex]x-k[\/latex].<\/li>\r\n\t<li>To find [latex]f\\left(k\\right)[\/latex], determine the remainder of the polynomial [latex]f\\left(x\\right)[\/latex] when it is divided by [latex]x-k[\/latex].<\/li>\r\n\t<li><em>k<\/em>\u00a0is a zero of [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\r\n\t<li>Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.<\/li>\r\n\t<li>When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.<\/li>\r\n\t<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\r\n\t<li>Every polynomial function has at least one complex zero.<\/li>\r\n\t<li>Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>\u00a0is a complex number.<\/li>\r\n\t<li>Another way to find the [latex]x[\/latex]<em>-<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the [latex]x[\/latex]-axis.<\/li>\r\n\t<li>The multiplicity of a zero determines how the graph behaves at the [latex]x[\/latex]-intercept.\r\n\r\n<ul>\r\n\t<li>The graph of a polynomial will cross the [latex]x[\/latex]-axis at a zero with odd multiplicity.<\/li>\r\n\t<li>The graph of a polynomial will touch and bounce off the [latex]x[\/latex]-axis at a zero with even multiplicity.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\r\n\t<li>A polynomial function of degree [latex]n[\/latex] has at most [latex]n\u2013\u00a01[\/latex] turning points.<\/li>\r\n\t<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most [latex]n\u2013\u00a01[\/latex] turning points.<\/li>\r\n\t<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<p><strong>average rate of change<\/strong><\/p>\r\n<p>the slope of a line between two points on the graph of a function, calculated via a ratio of the change in function output over the corresponding change in function input<\/p>\r\n<p><strong>continuous function<\/strong><\/p>\r\n<p>a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/p>\r\n<p><strong>degree of a polynomial<\/strong><\/p>\r\n<p>highest power of the variable that occurs in the polynomial<\/p>\r\n<p><strong>domain<\/strong><\/p>\r\n<p>the set of all input values into a function; the set of first components in the ordered pairs and each value in it is an input or independent variable, often labeled [latex]x[\/latex]<\/p>\r\n<p><strong>end behavior<\/strong><\/p>\r\n<p>the behavior of the graph of a function as the input decreases without bound and increases without bound<\/p>\r\n<p><strong>function<\/strong><\/p>\r\n<p>a relation in which each possible input value leads to exactly one output value<\/p>\r\n<p><strong>function notation<\/strong><\/p>\r\n<p>a notation used for representing output as a function of input, [latex]y=f(x)[\/latex], that is [latex]y[\/latex] is a function of [latex]x[\/latex]<\/p>\r\n<p><strong>graph transformation<\/strong><\/p>\r\n<p>involves shifting, stretching, or flipping its shape to create a new representation<\/p>\r\n<p><strong>horizontal line test<\/strong><\/p>\r\n<p>a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once<\/p>\r\n<p><strong>input<\/strong><\/p>\r\n<p>each object or value in a domain that relates to another object or value by a relationship known as a function<\/p>\r\n<p><strong>leading coefficient<\/strong><\/p>\r\n<p>the coefficient of the leading term<\/p>\r\n<p><strong>leading term<\/strong><\/p>\r\n<p>the term containing the variable with the highest power<\/p>\r\n<p><strong>linear function<\/strong><\/p>\r\n<p>characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[\/latex]<\/p>\r\n<p><strong>modeling<\/strong><\/p>\r\n<p>the process of translating real-world problems into mathematical terms and solving them<\/p>\r\n<p><strong>multiplicity<\/strong><\/p>\r\n<p>the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity [latex]p[\/latex].<\/p>\r\n<p><strong>one-to-one function<\/strong><\/p>\r\n<p>a function in which each output value corresponds to exactly one input value<\/p>\r\n<p><strong>output<\/strong><\/p>\r\n<p>each object or value in the range that is produced when an input value is entered into a function<\/p>\r\n<p><strong>range<\/strong><\/p>\r\n<p>the set of all output values of a function; the set of second components in the ordered pairs and each value in the range is an output or dependent variable, often labeled [latex]y[\/latex]<\/p>\r\n<p><strong>Rational Zero Theorem<\/strong> <strong>relation<\/strong><\/p>\r\n<p>a set of ordered pairs<\/p>\r\n<p><strong>slope<\/strong><\/p>\r\n<p>the ratio of the change in output values to the change in input values; a measure of the steepness of a line<\/p>\r\n<p><strong>synthetic division<\/strong><\/p>\r\n<p>a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x \u2013 k[\/latex]<\/p>\r\n<p><strong>term of a polynomial function<\/strong><\/p>\r\n<p>[latex]{a}_{i}{x}^{i}[\/latex]<\/p>\r\n<p><strong>vertex<\/strong><\/p>\r\n<p>the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/p>\r\n<p><strong>vertical line test<\/strong><\/p>\r\n<p>determines if a relation is a function by checking that no vertical line intersects the graph more than once<\/p>\r\n<p><strong>[latex]x[\/latex]-intercept<\/strong><\/p>\r\n<p>value of [latex]x[\/latex] where [latex]f(x) = 0[\/latex]<\/p>\r\n<p><strong>[latex]y[\/latex]-intercept<\/strong><\/p>\r\n<p>the value of a function when the input value is zero; the point at which the graph crosses the horizontal axis; also known as initial value<\/p>\r\n<p><strong>zeros<\/strong><\/p>\r\n<p>the [latex]x[\/latex]-intercepts of a quadratic equation<\/p>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>axis of symmetry<\/strong><\/p>\r\n<p>[latex]x=-\\dfrac{b}{2a}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are coming from the general form of a quadratic function\u00a0<\/p>\r\n<p><strong>calculating slope<\/strong><\/p>\r\n<p>[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\Rightarrow \\dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[\/latex]<\/p>\r\n<p><strong>Division Algorithm<\/strong><\/p>\r\n<p>given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\r\n<p><strong>Factor Theorem<\/strong><\/p>\r\n<p>[latex]k[\/latex] is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex]<\/p>\r\n<p><strong>general form of a polynomial function<\/strong><\/p>\r\n<p>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\n<p><strong>general form of a quadratic function<\/strong><\/p>\r\n<p>[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\r\n<p><strong>point-slope form<\/strong><\/p>\r\n<p>[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/p>\r\n<p><strong>Remainder Theorem<\/strong><\/p>\r\n<p>if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex]<\/p>\r\n<p><strong>slope-intercept form<\/strong><\/p>\r\n<p>[latex]\\begin{array}{lll}\\text{Equation form}\\hfill &amp; y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill &amp; f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\r\n<p><strong>standard form (vertex form) of a quadratic function<\/strong><\/p>\r\n<p>[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\n<p><strong>the [latex]x[\/latex]-coordinate of the vertex<\/strong><\/p>\r\n<p>[latex]h=-\\dfrac{b}{2a}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are coming from the general form of a quadratic function\u00a0<\/p>\r\n<p><strong>the [latex]y[\/latex]-coordinate of the vertex<\/strong><\/p>\r\n<p>[latex]k=f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are coming from the general form of a quadratic function\u00a0<\/p>","rendered":"<h2>Essential Concepts<\/h2>\n<ul>\n<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\n<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right)[\/latex].<\/li>\n<li>In table form, a function can be represented by rows or columns that relate to input and output values.<\/li>\n<li>To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.<\/li>\n<li>To solve for a specific function value, we determine the input values that yield the specific output value.<\/li>\n<li>An algebraic form of a function can be written from an equation.<\/li>\n<li>Input and output values of a function can be identified from a table.<\/li>\n<li>Relating input values to output values on a graph is another way to evaluate a function.<\/li>\n<li>A function is one-to-one if each output value corresponds to only one input value.<\/li>\n<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\n<li>A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.<\/li>\n<li>Fundamental toolkit functions include linear, quadratic, polynomial, exponential, and logarithmic functions, among others. Each type of function has a distinct graph shape. For example, linear functions have straight-line graphs, quadratic functions have parabolic graphs, and exponential functions have curved graphs that increase or decrease rapidly.<\/li>\n<li>The ordered pairs given by a linear function represent points on a line.<\/li>\n<li>Linear functions can be represented in words, function notation, tabular form and graphical form.<\/li>\n<li>The rate of change of a linear function is also known as the slope.<\/li>\n<li>An equation in slope-intercept form of a line includes the slope and the initial value of the function.<\/li>\n<li>The initial value, or [latex]y[\/latex]-intercept, is the output value when the input of a linear function is zero. It is the [latex]y[\/latex]-value of the point where the line crosses the [latex]y[\/latex]-axis.<\/li>\n<li>An increasing linear function results in a graph that slants upward from left to right and has a positive slope.\u00a0A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line.<\/li>\n<li>Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.<\/li>\n<li>The slope of a linear function can be calculated by dividing the difference between [latex]y[\/latex]-values by the difference in corresponding [latex]x[\/latex]-values of any two points on the line.<\/li>\n<li>The slope and initial value can be determined given a graph or any two points on the line.<\/li>\n<li>Point-slope form is useful for finding the equation of a linear function when given the slope of a line and one point. It is also convenient for finding the equation of a linear function when given two points through which a line passes.<\/li>\n<li>The equation for a linear function can be written in slope-intercept form if the slope [latex]m[\/latex]\u00a0and initial value [latex]b[\/latex] are known.<\/li>\n<li>Linear functions may be graphed by plotting points or by using the <em>y<\/em>-intercept and slope.<\/li>\n<li>Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections.<\/li>\n<li>The <em>y<\/em>-intercept and slope of a line may be used to write the equation of a line. The <em>x<\/em>-intercept is the point at which the graph of a linear function crosses the <em>x<\/em>-axis.<\/li>\n<li>Horizontal lines are written in the form, [latex]f(x)=b[\/latex]. Vertical lines are written in the form, [latex]x=b[\/latex].<\/li>\n<li>Transformations involve shifting, stretching, or compressing the graph of a function in various ways. For linear functions, these transformations can include vertical shifts, and vertical stretches or compressions.\n<ul>\n<li>Vertical Shift: Changing the y-intercept [latex]b[\/latex] in [latex]f(x) = mx + b[\/latex] results in a vertical shift of the line. If [latex]b[\/latex] increases, the line shifts up; if [latex]b[\/latex] decreases, it shifts down.<\/li>\n<li>Vertical Stretch or Compression: Altering the slope [latex]m[\/latex] affects the steepness of the line. A larger absolute value of [latex]m[\/latex] indicates a steeper slope. A positive [latex]m[\/latex] indicates an upward slope, while a negative [latex]m[\/latex] indicates a downward slope.<\/li>\n<\/ul>\n<\/li>\n<li>A polynomial function of degree two is called a quadratic function.<\/li>\n<li>The axis of symmetry is the vertical line passing through the vertex.<\/li>\n<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>The domain of a quadratic function is all real numbers. The range varies with the function.\n<ul>\n<li>If the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex].<\/li>\n<li>If the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.\n<ul>\n<li>The parabola opens upward if [latex]a > 0[\/latex] and downward if [latex]a < 0[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Transformations of quadratic functions\n<ul>\n<li>Vertical Shift General Form: Changing the constant [latex]c[\/latex] in the equation results in a vertical shift of the parabola. If [latex]c[\/latex] increases, the parabola shifts upwards; if [latex]c[\/latex] decreases, it shifts downwards.<\/li>\n<li>Vertical Shift Standard (Vertex) Form: Changing [latex]k[\/latex] results in a vertical shift of the entire parabola. Increasing [latex]k[\/latex] shifts the parabola upwards, while decreasing [latex]k[\/latex] shifts it downwards.<\/li>\n<li>Horizontal Shift: This can be achieved by substituting [latex]x[\/latex] with [latex](x - h)[\/latex], resulting in a function of the form [latex]f(x) = a(x - h)^2 + k[\/latex]. The parabola shifts to the right if [latex]h[\/latex] is positive and to the left if [latex]h[\/latex] is negative.<\/li>\n<li>Stretch\/Compression: Changing the value of [latex]a[\/latex] causes a vertical stretch or compression. If the absolute value of [latex]a[\/latex] is greater than 1, the parabola becomes narrower (stretches), and if it&#8217;s between 0 and 1, the parabola becomes wider (compresses).<\/li>\n<\/ul>\n<\/li>\n<li>A polynomial function is one whose equation contains only non-negative integer powers on the variable.<\/li>\n<li>The polynomial term containing the highest power on the variable is the leading term, and its degree is the number of the power. The leading coefficient of a polynomial is the coefficient of the leading term.<\/li>\n<li>The graph of a polynomial function describes a smooth, continuous curve.<\/li>\n<li>The domain of all polynomial functions is all real numbers.<\/li>\n<li>Even degree polynomial functions describe graphs whose ends both point up or both point down.<\/li>\n<li>Odd degree polynomial functions describe graphs whose ends points in opposite directions.<\/li>\n<li>The sign of the leading term will determine the direction of the ends of the graph:\n<ul>\n<li>even degree and positive coefficient: both ends point up<\/li>\n<li>even degree and negative coefficient: both ends point down<\/li>\n<li>odd degree and positive coefficient: the left-most end points down and the right-most end points up.<\/li>\n<li>odd degree and negative coefficient: the left-most end points up and the right-most end points down.<\/li>\n<\/ul>\n<\/li>\n<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\n<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\n<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form [latex]x-k[\/latex].<\/li>\n<li>To find [latex]f\\left(k\\right)[\/latex], determine the remainder of the polynomial [latex]f\\left(x\\right)[\/latex] when it is divided by [latex]x-k[\/latex].<\/li>\n<li><em>k<\/em>\u00a0is a zero of [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/li>\n<li>Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.<\/li>\n<li>When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.<\/li>\n<li>Synthetic division can be used to find the zeros of a polynomial function.<\/li>\n<li>Every polynomial function has at least one complex zero.<\/li>\n<li>Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\\left(x-c\\right)[\/latex] where <em>c<\/em>\u00a0is a complex number.<\/li>\n<li>Another way to find the [latex]x[\/latex]<em>&#8211;<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the [latex]x[\/latex]-axis.<\/li>\n<li>The multiplicity of a zero determines how the graph behaves at the [latex]x[\/latex]-intercept.\n<ul>\n<li>The graph of a polynomial will cross the [latex]x[\/latex]-axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch and bounce off the [latex]x[\/latex]-axis at a zero with even multiplicity.<\/li>\n<\/ul>\n<\/li>\n<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>A polynomial function of degree [latex]n[\/latex] has at most [latex]n\u2013\u00a01[\/latex] turning points.<\/li>\n<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most [latex]n\u2013\u00a01[\/latex] turning points.<\/li>\n<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>average rate of change<\/strong><\/p>\n<p>the slope of a line between two points on the graph of a function, calculated via a ratio of the change in function output over the corresponding change in function input<\/p>\n<p><strong>continuous function<\/strong><\/p>\n<p>a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/p>\n<p><strong>degree of a polynomial<\/strong><\/p>\n<p>highest power of the variable that occurs in the polynomial<\/p>\n<p><strong>domain<\/strong><\/p>\n<p>the set of all input values into a function; the set of first components in the ordered pairs and each value in it is an input or independent variable, often labeled [latex]x[\/latex]<\/p>\n<p><strong>end behavior<\/strong><\/p>\n<p>the behavior of the graph of a function as the input decreases without bound and increases without bound<\/p>\n<p><strong>function<\/strong><\/p>\n<p>a relation in which each possible input value leads to exactly one output value<\/p>\n<p><strong>function notation<\/strong><\/p>\n<p>a notation used for representing output as a function of input, [latex]y=f(x)[\/latex], that is [latex]y[\/latex] is a function of [latex]x[\/latex]<\/p>\n<p><strong>graph transformation<\/strong><\/p>\n<p>involves shifting, stretching, or flipping its shape to create a new representation<\/p>\n<p><strong>horizontal line test<\/strong><\/p>\n<p>a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once<\/p>\n<p><strong>input<\/strong><\/p>\n<p>each object or value in a domain that relates to another object or value by a relationship known as a function<\/p>\n<p><strong>leading coefficient<\/strong><\/p>\n<p>the coefficient of the leading term<\/p>\n<p><strong>leading term<\/strong><\/p>\n<p>the term containing the variable with the highest power<\/p>\n<p><strong>linear function<\/strong><\/p>\n<p>characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[\/latex]<\/p>\n<p><strong>modeling<\/strong><\/p>\n<p>the process of translating real-world problems into mathematical terms and solving them<\/p>\n<p><strong>multiplicity<\/strong><\/p>\n<p>the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity [latex]p[\/latex].<\/p>\n<p><strong>one-to-one function<\/strong><\/p>\n<p>a function in which each output value corresponds to exactly one input value<\/p>\n<p><strong>output<\/strong><\/p>\n<p>each object or value in the range that is produced when an input value is entered into a function<\/p>\n<p><strong>range<\/strong><\/p>\n<p>the set of all output values of a function; the set of second components in the ordered pairs and each value in the range is an output or dependent variable, often labeled [latex]y[\/latex]<\/p>\n<p><strong>Rational Zero Theorem<\/strong> <strong>relation<\/strong><\/p>\n<p>a set of ordered pairs<\/p>\n<p><strong>slope<\/strong><\/p>\n<p>the ratio of the change in output values to the change in input values; a measure of the steepness of a line<\/p>\n<p><strong>synthetic division<\/strong><\/p>\n<p>a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x \u2013 k[\/latex]<\/p>\n<p><strong>term of a polynomial function<\/strong><\/p>\n<p>[latex]{a}_{i}{x}^{i}[\/latex]<\/p>\n<p><strong>vertex<\/strong><\/p>\n<p>the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/p>\n<p><strong>vertical line test<\/strong><\/p>\n<p>determines if a relation is a function by checking that no vertical line intersects the graph more than once<\/p>\n<p><strong>[latex]x[\/latex]-intercept<\/strong><\/p>\n<p>value of [latex]x[\/latex] where [latex]f(x) = 0[\/latex]<\/p>\n<p><strong>[latex]y[\/latex]-intercept<\/strong><\/p>\n<p>the value of a function when the input value is zero; the point at which the graph crosses the horizontal axis; also known as initial value<\/p>\n<p><strong>zeros<\/strong><\/p>\n<p>the [latex]x[\/latex]-intercepts of a quadratic equation<\/p>\n<h2>Key Equations<\/h2>\n<p><strong>axis of symmetry<\/strong><\/p>\n<p>[latex]x=-\\dfrac{b}{2a}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are coming from the general form of a quadratic function\u00a0<\/p>\n<p><strong>calculating slope<\/strong><\/p>\n<p>[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\Rightarrow \\dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[\/latex]<\/p>\n<p><strong>Division Algorithm<\/strong><\/p>\n<p>given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\n<p><strong>Factor Theorem<\/strong><\/p>\n<p>[latex]k[\/latex] is a zero of polynomial function [latex]f\\left(x\\right)[\/latex] if and only if [latex]\\left(x-k\\right)[\/latex] \u00a0is a factor of [latex]f\\left(x\\right)[\/latex]<\/p>\n<p><strong>general form of a polynomial function<\/strong><\/p>\n<p>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p><strong>general form of a quadratic function<\/strong><\/p>\n<p>[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\n<p><strong>point-slope form<\/strong><\/p>\n<p>[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/p>\n<p><strong>Remainder Theorem<\/strong><\/p>\n<p>if a polynomial [latex]f\\left(x\\right)[\/latex] is divided by [latex]x-k[\/latex] , then the remainder is equal to the value [latex]f\\left(k\\right)[\/latex]<\/p>\n<p><strong>slope-intercept form<\/strong><\/p>\n<p>[latex]\\begin{array}{lll}\\text{Equation form}\\hfill & y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill & f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\n<p><strong>standard form (vertex form) of a quadratic function<\/strong><\/p>\n<p>[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p><strong>the [latex]x[\/latex]-coordinate of the vertex<\/strong><\/p>\n<p>[latex]h=-\\dfrac{b}{2a}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are coming from the general form of a quadratic function\u00a0<\/p>\n<p><strong>the [latex]y[\/latex]-coordinate of the vertex<\/strong><\/p>\n<p>[latex]k=f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are coming from the general form of a quadratic function\u00a0<\/p>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":72,"module-header":"cheat_sheet","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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8420"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8420\/revisions"}],"predecessor-version":[{"id":11698,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8420\/revisions\/11698"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/72"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8420\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8420"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8420"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8420"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}