Quadratic Functions

3. Explain why the condition of [latex]a\ne 0[/latex] is imposed in the definition of the quadratic function.

5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

7. [latex]g\left(x\right)={x}^{2}+2x - 3[/latex]

11. [latex]k\left(x\right)=3{x}^{2}-6x - 9[/latex]

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.

15. [latex]f\left(x\right)=2{x}^{2}-10x+4[/latex]

17. [latex]f\left(x\right)=4{x}^{2}+x - 1[/latex]

For the following exercises, determine the domain and range of the quadratic function.

21. [latex]f\left(x\right)={\left(x - 3\right)}^{2}+2[/latex]

23. [latex]f\left(x\right)={x}^{2}+6x+4[/latex]

For the following exercises, solve the equations over the complex numbers.

29. [latex]{x}^{2}+27=0[/latex]

31. [latex]{x}^{2}-4x+5=0[/latex]

35. [latex]{x}^{2}-10x+26=0[/latex]

39. [latex]2{x}^{2}+2x+5=0[/latex]

41. [latex]5{x}^{2}+6x+2=0[/latex]

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

53. [latex]f\left(x\right)={x}^{2}-2x[/latex]

55. [latex]f\left(x\right)={x}^{2}-5x - 6[/latex]

57. [latex]f\left(x\right)=-2{x}^{2}+5x - 8[/latex]

For the following exercises, write the equation for the graphed function.

59.

61.

63.

For the following exercises, use the table of values that represent points on the graph of a quadratic function. Find the general form of the equation of the quadratic function.

65.

x	–2	–1	0	1	2
y	5	2	1	2	5

67.

x	–2	–1	0	1	2
y	–2	1	2	1	–2

69.

x	–2	–1	0	1	2
y	8	2	0	2	8

 

85. Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

91. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by [latex]h\left(t\right)=-4.9{t}^{2}+229t+234[/latex]. Find the maximum height the rocket attains.

93. A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

Polynomial Functions

1. Explain the difference between the coefficient of a power function and its degree.

3. Explain the end behavior of a power function with odd degree if the leading coefficient is positive.

For the following exercises, find the degree and leading coefficient for the given polynomial.

13. [latex]7 - 2{x}^{2}[/latex]

15. [latex]x\left(4-{x}^{2}\right)\left(2x+1\right)[/latex]

For the following exercises, determine the end behavior of the functions.

17. [latex]f\left(x\right)={x}^{4}[/latex]

21. [latex]f\left(x\right)=-2{x}^{4}- 3{x}^{2}+ x - 1[/latex]

23. [latex]f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)[/latex]

For the following exercises, find the intercepts of the functions.

25. [latex]f\left(t\right)=2\left(t - 1\right)\left(t+2\right)\left(t - 3\right)[/latex]

27. [latex]f\left(x\right)={x}^{4}-16[/latex]

29. [latex]f\left(x\right)=x\left({x}^{2}-2x - 8\right)[/latex]

For the following exercises, determine the least possible degree of the polynomial function shown.

31.

33.

For the following exercises, describe the end behavior of the function.

47. [latex]f\left(x\right)={x}^{4}-5{x}^{2}[/latex]

49. [latex]f\left(x\right)=\left(x - 1\right)\left(x - 2\right)\left(3-x\right)[/latex]

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

51. [latex]f\left(x\right)={x}^{3}\left(x - 2\right)[/latex]

57. [latex]f\left(x\right)={x}^{4}-81[/latex]

59. [latex]f\left(x\right)={x}^{3}-2{x}^{2}-15x[/latex]

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

61. The y-intercept is [latex]\left(0,-4\right)[/latex]. The x-intercepts are [latex]\left(-2,0\right),\left(2,0\right)[/latex]. Degree is 2.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

63. The y-intercept is [latex]\left(0,0\right)[/latex]. The x-intercepts are [latex]\left(0,0\right),\left(2,0\right)[/latex]. Degree is 3.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

65. The y-intercept is [latex]\left(0,1\right)[/latex]. There is no x-intercept. Degree is 4.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

69. An open box is to be constructed by cutting out square corners of x-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of x.

Graphs of Polynomial Functions

For the following exercises, find the x– or t-intercepts of the polynomial functions.

7. [latex]C\left(t\right)=3\left(t+2\right)\left(t - 3\right)\left(t+5\right)[/latex]

9. [latex]C\left(t\right)=2t\left(t - 3\right){\left(t+1\right)}^{2}[/latex]

13. [latex]f\left(x\right)={x}^{3}+{x}^{2}-20x[/latex]

17. [latex]f\left(x\right)=2{x}^{3}-{x}^{2}-8x+4[/latex]

19. [latex]f\left(x\right)=2{x}^{4}+6{x}^{2}-8[/latex]

21. [latex]f\left(x\right)={x}^{6}-2{x}^{4}-3{x}^{2}[/latex]

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

25. [latex]f\left(x\right)={x}^{3}-9x[/latex], between [latex]x=2[/latex] and [latex]x=4[/latex].

27. [latex]f\left(x\right)=-{x}^{4}+4[/latex], between [latex]x=1[/latex] and [latex]x=3[/latex].

For the following exercises, find the zeros and give the multiplicity of each.

31. [latex]f\left(x\right)={x}^{2}{\left(2x+3\right)}^{5}{\left(x - 4\right)}^{2}[/latex]

33. [latex]f\left(x\right)={x}^{2}\left({x}^{2}+4x+4\right)[/latex]

37. [latex]f\left(x\right)={x}^{6}-{x}^{5}-2{x}^{4}[/latex]

39. [latex]f\left(x\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}[/latex]

For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior.

43. [latex]g\left(x\right)=\left(x+4\right){\left(x - 1\right)}^{2}[/latex]

45. [latex]k\left(x\right)={\left(x - 3\right)}^{3}{\left(x - 2\right)}^{2}[/latex]

47. [latex]n\left(x\right)=-3x\left(x+2\right)\left(x - 4\right)[/latex]

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

49.

53.

55.

For the following exercises, use the given information about the polynomial graph to write the equation.

59. Degree 5. Roots of multiplicity 2 at [latex]x=3[/latex] and [latex]x=1[/latex], and a root of multiplicity 1 at [latex]x=-3[/latex]. y-intercept at [latex]\left(0,9\right)[/latex]

63. Degree 3. Zeros at [latex]x=-3[/latex], [latex]x=-2[/latex] and [latex]x=1[/latex]. y-intercept at [latex]\left(0,12\right)[/latex].

65. Degree 4. Roots of multiplicity 2 at [latex]x=\frac{1}{2}[/latex] and roots of multiplicity 1 at [latex]x=6[/latex] and [latex]x=-2[/latex]. y-intercept at [latex]\left(0,18\right)[/latex].

For the following exercises, write the polynomial function that models the given situation.

75. A rectangle has a length of 10 units and a width of 8 units. Squares of x by x units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x.

77. A square has sides of 12 units. Squares [latex]x+1[/latex] by [latex]x+1[/latex] units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of x.

79. A right circular cone has a radius of [latex]3x+6[/latex] and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is [latex]V=\frac{1}{3}\pi {r}^{2}h[/latex] for radius r and height h.