?In Exercises 35–36, use the Rational Zero Theorem to list all possible rational zeros

In Exercises 1–4, graph the given quadratic function. Give each function’s domain and range. \(f(x)=(x-3)^{2}-4\) ________________ Equation Transcription: Text Transcription: f(x) = (x -3)^2 -4

In Exercises 1–4, graph the given quadratic function. Give each function’s domain and range. \(f(x)=5-(x+2)^{2}\) ________________ Equation Transcription: Text Transcription: f(x) = 5 -(x + 2)^2

In Exercises 1–4, graph the given quadratic function. Give each function’s domain and range. \(f(x)=-x^{2}-4 x+5\) ________________ Equation Transcription: Text Transcription: f(x) = -x^2 -4x + 5

In Exercises 1–4, graph the given quadratic function. Give each function’s domain and range. \(f(x)=3 x^{2}-6 x+1\) ________________ Equation Transcription: Text Transcription: f(x) = 3x^2 -6x + 1

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=(x-2)^{2}(x+1)^{3}\) ________________ Equation Transcription: Text Transcription: f(x) = (x -2)^2(x + 1)^3

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=-(x-2)^{2}(x+1)\) ________________ Equation Transcription: Text Transcription: f(x) = -(x -2)^2(x + 1)

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=x^{3}-x^{2}-4 x+4\) ________________ Equation Transcription: Text Transcription: f(x) = x^3 -x^2 -4x + 4

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=x^{4}-5 x^{2}+4\) ________________ Equation Transcription: Text Transcription: f(x) = x^4 -5x^2 + 4

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=-(x+1)^{6}\) ________________ Equation Transcription: Text Transcription: f(x) = -(x + 1)^6

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=-6 x^{3}+7 x^{2}-1\) ________________ Equation Transcription: Text Transcription: f(x) = -6x^3 + 7x^2 -1

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=2 x^{3}-2 x\) ________________ Equation Transcription: Text Transcription: f(x) = 2x^3 -2x

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=x^{3}-2 x^{2}+26 x\) ________________ Equation Transcription: Text Transcription: f(x) = x^3 -2x^2 + 26x

In Exercises 5–13, find all zeros of each polynomial function. Then graph the function. \(f(x)=-x^{3}+5 x^{2}-5 x-3\) ________________ Equation Transcription: Text Transcription: f(x) = -x^3 + 5x^2 -5x -3

In Exercises 14–19, solve each polynomial equation \(x^{3}-3 x+2=0\) ________________ Equation Transcription: Text Transcription: x^3 -3x + 2 = 0

In Exercises 14–19, solve each polynomial equation \(6x^{3}-11 x^{2}+6 x-1=0\) ________________ Equation Transcription: Text Transcription: 6x^3 -11x^2 + 6x -1 = 0

In Exercises 14–19, solve each polynomial equation \((2 x+1)(3 x-2)^{3}(2 x-7)=0\) ________________ Equation Transcription: Text Transcription: (2 x + 1)(3x -2)^3(2x -7) = 0

In Exercises 14–19, solve each polynomial equation \(2 x^{3}+5 x^{2}-200 x-500=0\) ________________ Equation Transcription: Text Transcription: 2x^3 + 5x^2 -200x -500 = 0

In Exercises 14–19, solve each polynomial equation \(x^{4}-x^{3}-11 x^{2}=x+12\) ________________ Equation Transcription: Text Transcription: x^4 -x^3 -11x^2 = x + 12

In Exercises 14–19, solve each polynomial equation \(2 x^{4}+x^{3}-17 x^{2}-4 x+6=0\) ________________ Equation Transcription: Text Transcription: 2x^4 + x^3 -17x^2 -4x + 6 = 0

A company manufactures and sells bath cabinets. The function \(P(x)=-x^{2}+150 x-4425\) models the company’s daily profit, P(x), when x cabinets are manufactured and sold per day. How many cabinets should be manufactured and sold per day to maximize the company’s profit? What is the maximum daily profit? ________________ Equation Transcription: Text Transcription: P(x) = -x^2 + 150x -4425

Among all pairs of numbers whose sum is -18, find a pair whose product is as large as possible. What is the maximum product?

The base of a triangle measures 40 inches minus twice the measure of its height. For what measure of the height does the triangle have a maximum area? What is the maximum area?

In Exercises 23–24, divide, using synthetic division if possible. \(\left(6 x^{4}-3 x^{3}-11 x^{2}+2 x+4\right) \div\left(3 x^{2}-1\right)\) ________________ Equation Transcription: Text Transcription: (6x^4 -3x^3 -11x^2 + 2x + 4) div (3x^2 -1)

In Exercises 23–24, divide, using synthetic division if possible. \(\left(2 x^{4}-13 x^{3}+17 x^{2}+18 x-24\right) \div(x-4)\) ________________ Equation Transcription: Text Transcription: (2x^4 -13x^3 + 17x^2 + 18x -24) div (x -4)

In Exercises 25–26, find an nth-degree polynomial function with real coefficients satisfying the given conditions. \(n=3 ; 1 \text { and i are zeros; } f(-1)=8\) ________________ Equation Transcription: Text Transcription: n = 3; 1 and i are zeros; f(-1) = 8

In Exercises 25–26, find an nth-degree polynomial function with real coefficients satisfying the given conditions. \(n=4 ; 2 \text { (with multiplicity 2) and 3i are zeros; } f(0)=36\) ________________ Equation Transcription: Text Transcription: n = 4; 2 (with multiplicity 2) and 3i are zeros; f(0) = 36

Does \(f(x)=x^{3}-x-5\) have a real zero between 1 and 2? ________________ Equation Transcription: Text Transcription: f(x) = x^3 -x -5

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. \(f(x)=-(x+1)^{2}+4\) ________________ Equation Transcription: Text Transcription: f(x)=-(x+1)^2+4

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. \(f(x)=-(x+4)^{2}-2\) ________________ Equation Transcription: Text Transcription: f(x)=-(x+4)^2-2

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. \(f(x)=-x^{2}+2 x+3\) ________________ Equation Transcription: Text Transcription: f(x)=-x^2+2x+3

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. \(f(x)=2 x^{2}-4 x-6\) ________________ Equation Transcription: Text Transcription: f(x)=2x^2-4x-6

In Exercises 5–6, use the function’s equation, and not its graph, to find a. the minimum or maximum value and where it occurs. b. the function’s domain and its range \(f(x)=-x^{2}+14 x-106\) ________________ Equation Transcription: Text Transcription: f(x)=-x^2+14x-106

In Exercises 5–6, use the function’s equation, and not its graph, to find a. the minimum or maximum value and where it occurs. b. the function’s domain and its range \(f(x)=2 x^{2}+12 x+703\) ________________ Equation Transcription: Text Transcription: f(x)=2x^2+12x+703

A quarterback tosses a football to a receiver 40 yards downfield. The height of the football, f(x), in feet, can be modeled by \(f(x)=-0.025 x^{2}+x+6\), where x is the ball’s horizontal distance, in yards, from the quarterback. a. What is the ball’s maximum height and how far from the quarterback does this occur? b. From what height did the quarterback toss the football? c. If the football is not blocked by a defensive player nor caught by the receiver, how far down the field will it go before hitting the ground? Round to the nearest tenth of a yard. d. Graph the function that models the football’s parabolic path. ________________ Equation Transcription: Text Transcription: f(x) = -0.025x^2 + x + 6

A field bordering a straight stream is to be enclosed. The side bordering the stream is not to be fenced. If 1000 yards of fencing material is to be used, what are the dimensions of the largest rectangular field that can be fenced? What is the maximum area

Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ] \(f(x)=-x^{3}+x^{2}+2 x\) ________________ Equation Transcription: Text Transcription: f(x)=-x^3+x^2+2x

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ] \(f(x)=x^{6}-6 x^{4}+9 x^{2}\) ________________ Equation Transcription: Text Transcription: f(x)=x^6-6x^4+9x^2

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ] \(f(x)=x^{5}-5 x^{3}+4 x\) ________________ Equation Transcription: Text Transcription: f(x)=x^5-5x^3+4x

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ] \(f(x)=-x^{4}+1\) ________________ Equation Transcription: Text Transcription: f(x)=-x^4+1

The Brazilian Amazon rain forest is the world’s largest tropical rain forest, with some of the greatest biodiversity of any region. In 2009, the number of trees cut down in the Amazon dropped to its lowest level in 20 years. The line graph shows the number of square kilometers cleared from 2001 through 2009. The data in the line graph can be modeled by the following third- and fourth-degree polynomial functions: a. Use the Leading Coefficient Test to determine the end behavior to the right for the graph of f . b. Assume that the rate at which the Amazon rain forest is being cut down continues to decline. Based on your answer to part (a), will f be useful in modeling Amazon deforestation over an extended period of time? Explain your answer. c. Use the Leading Coefficient Test to determine the end behavior to the right for the graph of g d. Assume that the rate at which the Amazon rain forest is being cut down continues to decline. Based on your answer to part (c), will g be useful in modeling Amazon deforestation over an extended period of time? Explain your answer.

The polynomial function \(f(x)=-0.87 x^{3}+0.35 x^{2}+81.62 x+7684.94\) model the number of thefts, f(x), in thousands in the United States x years after 1987. Will this function be useful in modeling the number of thefts over an extended period of time? Explain your answer. Equation Transcription: Text Transcription: f(x)=-0.87x^3 +0.35x^2 +81.62x +7684.94

In Exercises 16–17, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. \(f(x)=-2(x-1)(x+2)^{2}(x+5)^{3}\) Equation Transcription: Text Transcription: f(x)=-2(x-1)(x+2)^2 (x+5)^3

In Exercises 16–17, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. \(f(x)=x^{3}-5 x^{2}-25 x+125\) Equation Transcription: Text Transcription: f(x)=x^3 -5x^2 -25x+125

Show that has a real zero between 1 and 2. \(f(x)=x^{3}-2 x-1\) Equation Transcription: Text Transcription: f(x)=x^3 -2x-1

In Exercises 19–24, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. c. Graph the function. \(f(x)=x^{3}-x^{2}-9 x+9\) Equation Transcription: Text Transcription: f(x)=x^3 -x^2 -9x+9

In Exercises 19–24, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. c. Graph the function. \(f(x)=4 x-x^{3}\) Equation Transcription: Text Transcription: f(x)=4x-x^3

In Exercises 19–24, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. c. Graph the function. \(f(x)=2 x^{3}+3 x^{2}-8 x-12\) Equation Transcription: Text Transcription: f(x)=2x^3 +3x^2 -8x-12

In Exercises 19–24, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. c. Graph the function. \(f(x)=-x^{4}+25 x^{2}\) Equation Transcription: Text Transcription: f(x)=-x^4 +25x^2

In Exercises 19–24, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. c. Graph the function. \(f(x)=-x^{4}+6 x^{3}-9 x^{2}\) Equation Transcription: Text Transcription: f(x)=-x^4 +6x^3 -9x^2

In Exercises 19–24, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. c. Graph the function. \(f(x)=3 x^{4}-15 x^{3}\) Equation Transcription: Text Transcription: f(x)=3x^4 -15x^3

In Exercises 25–26, graph each polynomial function. \(f(x)=2 x^{2}(x-1)^{3}(x+2)\) Equation Transcription: Text Transcription: f(x)=2x^2 (x-1)^3 (x+2)

In Exercises 25–26, graph each polynomial function. \(f(x)=-x^{3}(x+4)^{2}(x-1)\) Equation Transcription: Text Transcription: f(x)=-x^3 (x+4)^2 (x-1)

In Exercises 27–29, divide using long division. \(\left(4 x^{3}-3 x^{2}-2 x+1\right) \div(x+1)\) Equation Transcription: Text Transcription: (4x^3 -3x^2 -2x+1) div (x+1)

In Exercises 27–29, divide using long division. \(\left(10 x^{3}-26 x^{2}+17 x-13\right) \div(5 x-3)\) Equation Transcription: Text Transcription: (10x^3 -26x^2 +17x-13) div (5x-3)

In Exercises 27–29, divide using long division. \(\left(4 x^{4}+6 x^{3}+3 x-1\right) \div\left(2 x^{2}+1\right)\) Equation Transcription: Text Transcription: (4x^4 +6x^3 +3x-1) div (2x^2 +1)

In Exercises 30–31, divide using synthetic division. \(\left(3 x^{4}+11 x^{3}-20 x^{2}+7 x+35\right) \div(x+5)\) Equation Transcription: Text Transcription: (3x^4 +11x^3 -20x^2 +7x+35) div (x+5)

In Exercises 30–31, divide using synthetic division. \(\left(3 x^{4}-2 x^{2}-10 x\right) \div(x-2)\) Equation Transcription: Text Transcription: (3x^4 -2x^2 -10x) div (x-2)

Given \(f(x)=2 x^{3}-7 x^{2}+9 x-3\), use the Remainder Theorem to find f(-13). Equation Transcription: Text Transcription: f(x)=2x^3 -7x^2 +9x-3

Use synthetic division to divide by x - 2. Use the result to find all zeros of f. \(f(x)=2 x^{3}+x^{2}-13 x+6\) Equation Transcription: Text Transcription: f(x)=2x^3 +x^2 -13x+6

Solve the equation \(x^{3}-17 x+4=0\) given that 4 is a root. Equation Transcription: Text Transcription: x^3 -17x+4=0

In Exercises 35–36, use the Rational Zero Theorem to list all possible rational zeros for each given function. \(f(x)=x^{4}-6 x^{3}+14 x^{2}-14 x+5\) Equation Transcription: Text Transcription: f(x)=x^4 -6x^3 +14x^2 -14x+5

In Exercises 35–36, use the Rational Zero Theorem to list all possible rational zeros for each given function. \(f(x)=3 x^{5}-2 x^{4}-15 x^{3}+10 x^{2}+12 x-8\) Equation Transcription: Text Transcription: f(x)=3x^5 -2x^4 -15x^3 +10x^2 +12x-8

In Exercises 37–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. \(f(x)=3 x^{4}-2 x^{3}-8 x+5\) Equation Transcription: Text Transcription: f(x)=3x^4 -2x^3 -8x+5

In Exercises 37–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. \(f(x)=2 x^{5}-3 x^{3}-5 x^{2}+3 x-1\) Equation Transcription: Text Transcription: f(x)=2x^5 -3x^3 -5x^2 +3x-1

Use Descarte’s Rule of Signs to explain why \(2 x^{4}+6 x^{2}+8=0\) has no real roots. Equation Transcription: Text Transcription: 2x^4 +6x^2 +8=0

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(f(x)=x^{3}+3 x^{2}-4\) Equation Transcription: Text Transcription: f(x)=x^3 +3x^2 -4

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(f(x)=6 x^{3}+x^{2}-4 x+1\) Equation Transcription: Text Transcription: f(x)=6x^3 +x^2 -4x+1

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(8 x^{3}-36 x^{2}+46 x-15=0\) Equation Transcription: Text Transcription: 8x^3 -36x^2 +46x-15=0

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(2 x^{3}+9 x^{2}-7 x+1=0\) Equation Transcription: Text Transcription: 2x^3 +9x^2 -7x+1=0

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(x^{4}-x^{3}-7 x^{2}+x+6=0\) Equation Transcription: Text Transcription: x^4 -x^3 -7x^2 +x+6=0

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(4 x^{4}+7 x^{2}-2=0\) Equation Transcription: Text Transcription: 4x^4 +7x^2 -2=0

For Exercises 40–46, a. List all possible rational roots or rational zeros. b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. \(f(x)=2 x^{4}+x^{3}-9 x^{2}-4 x+4\) Equation Transcription: Text Transcription: f(x)=2x^4 +x^3 -9x^2 -4x+4

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, graph the function and verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, graph the function and verify the real zeros and the given function value. n = 4; i is a zero; -3 is a zero of multiplicity 2; f(-1) = 16

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. \(f(x)=2 x^{4}+3 x^{3}+3 x-2\) Equation Transcription: Text Transcription: f(x)=2x^4 +3x^3 +3x-2

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. \(g(x)=x^{4}-6 x^{3}+x^{2}+24 x+16\) Equation Transcription: Text Transcription: g(x)=x^4 -6x^3 +x^2 +24x+16

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

In Exercises 55–56, use transformations of \(f(x)=\frac{1}{x} \text { or } f(x)=\frac{1}{x^{2}}\) to graph each rational function. \(g(x)=\frac{1}{(x+2)^{2}}-1\) Equation Transcription: Text Transcription: f(x)=1/x or f(x)=1/x^2 g(x)=1/(x+2)^2 -1

In Exercises 55–56, use transformations of \(f(x)=\frac{1}{x} \text { or } f(x)=\frac{1}{x^{2}}\) to graph each rational function. \(h(x)=\frac{1}{x-1}+3\) Equation Transcription: Text Transcription: f(x)=1/x or f(x)=1/x^2 h(x)=1/x-1 +3

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(f(x)=\frac{2 x}{x^{2}-9}\) Equation Transcription: Text Transcription: f(x)=2x/x^2 -9

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(g(x)=\frac{2 x-4}{x+3}\) Equation Transcription: Text Transcription: g(x)=2x-4/x+3

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(h(x)=\frac{x^{2}-3 x-4}{x^{2}-x-6}\) Equation Transcription: Text Transcription: h(x)=x^2 -3x-4/x^2 -x-6

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(r(x)=\frac{x^{2}+4 x+3}{(x+2)^{2}}\) Equation Transcription: Text Transcription: r(x)=x^2 +4x+3/(x+2)^2

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(y=\frac{x^{2}}{x+1}\) Equation Transcription: Text Transcription: y=x^2 /x+1

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(y=\frac{x^{2}+2 x-3}{x-3}\) Equation Transcription: Text Transcription: y=x^2 +2x-3/x-3

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(f(x)=\frac{-2 x^{3}}{x^{2}+1}\) Equation Transcription: Text Transcription: f(x)=-2x^3 /x^2 +1

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. \(g(x)=\frac{4 x^{2}-16 x+16}{2 x-3}\) Equation Transcription: Text Transcription: g(x)=4x^2 -16x+16/2x-3

A company is planning to manufacture affordable graphing calculators. The fixed monthly cost will be $50,000 and it will cost $25 to produce each calculator. a. Write the cost function, C, of producing x graphing calculators. b. Write the average cost function, \(\bar{C}\), of producing x graphing calculators. c. Find and interpret \(\bar{C}(50), \bar{C}(100), \bar{C}(1000), \text { and } \bar{C}(100,000)\). d. What is the horizontal asymptote for the graph of this function and what does it represent? Equation Transcription: Text Transcription: C overline C overline (50), C overline (100), C overline (1000), and C overline (100,000)

Exercises 66–67 involve rational functions that model the given situations. In each case, find the horizontal asymptote as x ? ? and then describe what this means in practical terms. \(f(x)=\frac{150 x+120}{0.05 x+1}\); the number of bass, f(x), after x months in a lake that was stocked with 120 bass Equation Transcription: Text Transcription: f(x)=150x+120/0.05x +1

Exercises 66–67 involve rational functions that model the given situations. In each case, find the horizontal asymptote as x ? ? and then describe what this means in practical terms. \(P(x)=\frac{72,900}{100 x^{2}+729}\); the percentage, P(x), of people in the United States with x years of education who are unemployed Equation Transcription: Text Transcription: P(x)=72, 900/100x^2 +729

The bar graph shows the population of the United States, in millions, for six selected years. Here are two functions that model the data: a. Write a function that models the total U.S. population, P(x), in millions, x years after 1985. b. Write a rational function that models the fraction of men in the U.S. population, R(x), x years after 1985. c. What is the equation of the horizontal asymptote associated with the function in part (b)? Round to two decimal places. What does this mean about the percentage of men in the U.S. population over time?

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. \(2 x^{2}+5 x-3<0\) Equation Transcription: Text Transcription: 2x^2 +5x-3 < 0

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. \(2 x^{2}+9 x+4 \geq 0\) Equation Transcription: Text Transcription: 2x^2 +9x+4 geq 0

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. \(x^{3}+2 x^{2}>3 x\) Equation Transcription: Text Transcription: x^3 +2x^2 >3x

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. \(\frac{x-6}{x+2}>0\) Equation Transcription: Text Transcription: x-6/x+2 >0

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. \(\frac{(x+1)(x-2)}{x-1} \geq 0\) Equation Transcription: Text Transcription: (x+1)(x-2)/x-1 geq 0

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. \(\frac{x+3}{x-4} \leq 5\) Equation Transcription: Text Transcription: x+3/x-4 leq 5

The graph shows stopping distances for motorcycles at various speeds on dry roads and on wet roads. The functions model a motorcycle’s stopping distance, f(x) or g(x), in feet, traveling at x miles per hour. Function f models stopping distance on dry pavement and function g models stopping distance on wet pavement. a. Use function g to find the stopping distance on wet pavement for a motorcycle traveling at 35 miles per hour. Round to the nearest foot. Does your rounded answer overestimate or underestimate the stopping distance shown by the graph? By how many feet? b. Use function f to determine speeds on dry pavement requiring stopping distances that exceed 267 feet.

Use the position function \(s(t)=-16 t^{2}+v_{0} t+s_{0}\) to solve this problem. A projectile is fired vertically upward from ground level with an initial velocity of 48 feet per second. During which time period will the projectile’s height exceed 32 feet? Equation Transcription: Text Transcription: s(t)=-16t^2 +v_0 t + s_0

Solve the variation problems in Exercises 77–82. Many areas of Northern California depend on the snowpack of the Sierra Nevada mountain range for their water supply. The volume of water produced from melting snow varies directly as the volume of snow. Meteorologists have determined that 250 cubic centimeters of snow will melt to 28 cubic centimeters of water. How much water does 1200 cubic centimeters of melting snow produce?

Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?

Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

Solve the variation problems in Exercises 77–82. The loudness of a stereo speaker, measured in decibels, varies inversely as the square of your distance from the speaker. When you are 8 feet from the speaker, the loudness is 28 decibels. What is the loudness when you are 4 feet from the speaker?

Solve the variation problems in Exercises 77–82. The time required to assemble computers varies directly as the number of computers assembled and inversely as the number of workers. If 30 computers can be assembled by 6 workers in 10 hours, how long would it take 5 workers to assemble 40 computers?

Solve the variation problems in Exercises 77–82. The volume of a pyramid varies jointly as its height and the area of its base. A pyramid with a height of 15 feet and a base with an area of 35 square feet has a volume of 175 cubic feet. Find the volume of a pyramid with a height of 20 feet and a base with an area of 120 square feet.

Heart rates and life spans of most mammals can be modeled using inverse variation. The bar graph shows the average heart rate and the average lifespan of five mammals. a. A mammal’s average life span, L, in years, varies inversely as its average heart rate, R, in beats per minute. Use the data shown for horses to write the equation that models this relationship. b. Is the inverse variation equation in part (a) an exact model or an approximate model for the data shown for lions? c. Elephants have an average heart rate of 27 beats per minute. Determine their average life span.

In Exercises 1–2, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. \(f(x)=(x+1)^{2}+4\) Equation Transcription: Text Transcription: f(x)=(x+1)^2 +4

In Exercises 1–2, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range. \(f(x)=x^{2}-2 x-3\) Equation Transcription: Text Transcription: f(x)=x^2 -2x-3

Determine, without graphing, whether the quadratic function \(f(x)=-2 x^{2}+12 x-16\) has a minimum value or a maximum value. Then find a. the minimum or maximum value and where it occurs. b. the function’s domain and its range. Equation Transcription: Text Transcription: f(x)=-2x^2 +12x-16

The function \(f(x)=-x^{2}+46 x-360\) models the daily profit, f(x), in hundreds of dollars, for a company that manufactures x computers daily. How many computers should be manufactured each day to maximize profit? What is the maximum daily profit? Equation Transcription: Text Transcription: f(x)=-x^2 +46x-360

Among all pairs of numbers whose sum is 14, find a pair whose product is as large as possible. What is the maximum product?

Consider the function \(f(x)=x^{3}-5 x^{2}-4 x+20\). a. Use factoring to find all zeros of f. b. Use the Leading Coefficient Test and the zeros of f to graph the function. Equation Transcription: Text Transcription: f(x)=x^3 -5x^2 -4x+20

Use end behavior to explain why the following graph cannot be the graph of \(f(x)=x^{5}-x\) . Then use intercepts to explain why the graph cannot represent \(f(x)=x^{5}-x\) . Equation Transcription: Text Transcription: f(x)=x^5 -x

The graph of \(f(x)=6 x^{3}-19 x^{2}+16 x-4\) is shown in the figure. a. Based on the graph of f, find the root of the equation \(6 x^{3}-19 x^{2}+16 x-4=0\) that is an integer. b. Use synthetic division to find the other two roots of \(6 x^{3}-19 x^{2}+16 x-4=0\). Equation Transcription: Text Transcription: f(x)=6x^3 -19x^2 +16x-4 6x^3 -19x^2 +16x -4=0

Use the Rational Zero Theorem to list all possible rational zeros of \(f(x)=2 x^{3}+11 x^{2}-7 x-6\). Equation Transcription: Text Transcription: f(x)=2x^3 +11x^2 -7x-6

Use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros of \(f(x)=3 x^{5}-2 x^{4}-2 x^{2}+x-1\) Equation Transcription: Text Transcription: f(x)=3x^5 -2x^4 -2x^2 +x-1

Solve: \(x^{3}+9 x^{2}+16 x-6=0\). Equation Transcription: Text Transcription: x^3 +9x^2 +16x-6=0

Consider the function whose equation is given by \(f(x)=2 x^{4}-x^{3}-13 x^{2}+5 x+15\). a. List all possible rational zeros. b. Use the graph of f in the figure shown and synthetic division to find all zeros of the function. Equation Transcription: Text Transcription: f(x)=2x^4 -x^3 -13x^2 +5x+15

Use the graph of \(f(x)=x^{3}+3 x^{2}-4\) in the figure shown to factor \(x^{3}+3 x^{2}-4\). Equation Transcription: Text Transcription: f(x)=x^3 +3x^2 -4 x^3 +3x^2 -4

Find a fourth-degree polynomial function f(x) with real coefficients that has -1, 1, and i as zeros and such that f(3) = 160.

The figure shows an incomplete graph of \(f(x)=-3 x^{3}-4 x^{2}+x+2\) . Find all the zeros of the function. Then draw a complete graph. Equation Transcription: Text Transcription: f(x)=-3x^3 -4x^2 +x+2

In Exercises 16–21, find the domain of each rational function and graph the function. \(f(x)=\frac{1}{(x+3)^{2}}\) Equation Transcription: Text Transcription: f(x)=1/(x+3)^2

In Exercises 16–21, find the domain of each rational function and graph the function. \(f(x)=\frac{1}{x-1}+2\) Equation Transcription: Text Transcription: f(x)=1/x-1 +2

In Exercises 16–21, find the domain of each rational function and graph the function. \(f(x)=\frac{x}{x^{2}-16}\) Equation Transcription: Text Transcription: f(x)=x/x^2 -16

In Exercises 16–21, find the domain of each rational function and graph the function. \(f(x)=\frac{x^{2}-9}{x-2}\) Equation Transcription: Text Transcription: f(x)=x^2 -9/x-2

In Exercises 16–21, find the domain of each rational function and graph the function. \(f(x)=\frac{x+1}{x^{2}+2 x-3}\) Equation Transcription: Text Transcription: f(x)=x+1/x^2 +2x-3

In Exercises 16–21, find the domain of each rational function and graph the function. \(f(x)=\frac{4 x^{2}}{x^{2}+3}\) Equation Transcription: Text Transcription: f(x)=4x^2 /x^2 +3

A company is planning to manufacture portable satellite radio players. The fixed monthly cost will be $300,000 and it will cost $10 to produce each player. a. Write the average cost function, \(\bar{C}\), of producing x players. b. What is the horizontal asymptote for the graph of this function and what does it represent? Equation Transcription: Text Transcription: bar C

Solve each inequality in Exercises 23–24 and graph the solution set on a real number line. Express each solution set in interval notation. \(x^{2}<x+12\) Equation Transcription: Text Transcription: x^2 < x+12

Solve each inequality in Exercises 23–24 and graph the solution set on a real number line. Express each solution set in interval notation. \(\frac{2 x+1}{x-3} \leq 3\) Equation Transcription: Text Transcription: 2x+1/x-3 leq 3

The intensity of light received at a source varies inversely as the square of the distance from the source. A particular light has an intensity of 20 foot-candles at 15 feet. What is the light’s intensity at 10 feet?