🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Precalculus with Limits ## Ron Larson ## Chapter 2 ## Polynomial and Rational Functions ## Educators JH    + 2 more educators ### Problem 1 Fill in the blanks. Linear, constant, and squaring functions are examples of __________ functions. JH J H. Numerade Educator ### Problem 2 Fill in the blanks. A polynomial function of degree and leading coefficient$ a_n $is a function of the form$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 (a_n \neq 0) $where$ n $is a ________ _________ and$ a_n, a_{n-1}, \cdots , a_1, a_0 $are ________ numbers. JH J H. Numerade Educator ### Problem 3 Fill in the blanks. A __________ function is a second-degree polynomial function, and its graph is called a __________. JH J H. Numerade Educator ### Problem 4 Fill in the blanks. The graph of a quadratic function is symmetric about its ________. Suzanne W. Numerade Educator ### Problem 5 Fill in the blanks. If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________. JH J H. Numerade Educator ### Problem 6 Fill in the blanks. If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________. JH J H. Numerade Educator ### Problem 7 In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]$ f(x) = (x - 2)^2 $JH J H. Numerade Educator ### Problem 8 In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]$ f(x) = (x + 4)^2 $JH J H. Numerade Educator ### Problem 9 In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]$ f(x) = x^2 - 2 $JH J H. Numerade Educator ### Problem 10 In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]$ f(x) = (x + 1)^2 - 2 $JH J H. Numerade Educator ### Problem 11 In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]$ f(x) = 4 - (x - 2)^2 $JH J H. Numerade Educator ### Problem 12 In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]$ f(x) = -(x - 4)^2 $JH J H. Numerade Educator ### Problem 13 In Exercises 13-16, graph each function. Compare the graph of each function with the graph of$ y = x^2 $. (a)$ f(x) = \frac{1}{2} x^2 $(b)$ g(x) = -\frac{1}{8} x^2 $(c)$ h(x) = \frac{3}{2} x^2 $(d)$ k(x) = -3x^2 $JH J H. Numerade Educator ### Problem 14 In Exercises 13-16, graph each function. Compare the graph of each function with the graph of$ y = x^2 $. (a)$ f(x) = x^2 + 1 $(b)$ g(x) = x^2 - 1 $(c)$ h(x) = x^2 + 3 $(d)$ k(x) = x^2 - 3 $JH J H. Numerade Educator ### Problem 15 In Exercises 13-16, graph each function. Compare the graph of each function with the graph of$ y = x^2 $. (a)$ f(x) = (x - 1)^2 $(b)$ g(x) = (3x)^2 + 1 $(c)$ h(x) = \left(\frac{1}{3} x^2 \right) - 3 $(d)$ k(x) = (x + 3)^2 $JH J H. Numerade Educator ### Problem 16 In Exercises 13-16, graph each function. Compare the graph of each function with the graph of$ y = x^2 $. (a)$ f(x) = -\frac{1}{2} (x - 2)^2 + 1 $(b)$ g(x) = \left[\frac{1}{2} (x -1) \right]^2 - 3 $(c)$ h(x) = -\frac{1}{2} (x +1)^2 - 1 $(d)$ k(x) = [2(x + 1)]^2 +4 $ Suzanne W. Numerade Educator ### Problem 17 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = 1 - x^2 $JH J H. Numerade Educator ### Problem 18 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ g(x) = x^2 - 8 $JH J H. Numerade Educator ### Problem 19 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = x^2 + 7 $JH J H. Numerade Educator ### Problem 20 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ h(x) = 12 - x^2 $JH J H. Numerade Educator ### Problem 21 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = \frac{1}{2} x^2 - 4 $JH J H. Numerade Educator ### Problem 22 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = 16 - \frac{1}{4} x^2 $JH J H. Numerade Educator ### Problem 23 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = (x + 4)^2 - 3 $JH J H. Numerade Educator ### Problem 24 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = (x - 6)^2 + 8 $JH J H. Numerade Educator ### Problem 25 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ h(x) = x^2 - 8x + 16 $JH J H. Numerade Educator ### Problem 26 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ g(x) = x^2 + 2x + 1 $JH J H. Numerade Educator ### Problem 27 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = x^2 - x + \frac{5}{4} $JH J H. Numerade Educator ### Problem 28 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = x^2 + 3x + \frac{1}{4} $JH J H. Numerade Educator ### Problem 29 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = - x^2 + 2x + 5 $JH J H. Numerade Educator ### Problem 30 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = - x^2 - 4x + 1 $JH J H. Numerade Educator ### Problem 31 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ h(x) = 4x^2 - 4x + 21 $JH J H. Numerade Educator ### Problem 32 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = 2x^2 - x + 1 $JH J H. Numerade Educator ### Problem 33 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = \frac{1}{4} x^2 - 2x - 12 $JH J H. Numerade Educator ### Problem 34 In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).$ f(x) = -\frac{1}{3} x^2 + 3x - 6 $ Suzanne W. Numerade Educator ### Problem 35 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ f(x) = - (x^2 + 2x - 3) $JH J H. Numerade Educator ### Problem 36 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ f(x) = - (x^2 + x - 30) $JH J H. Numerade Educator ### Problem 37 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ g(x) = x^2 + 8x + 11 $JH J H. Numerade Educator ### Problem 38 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ f(x) = x^2 + 10x + 14 $ James K. Numerade Educator ### Problem 39 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ f(x) = 2x^2 - 16x + 31 $JH J H. Numerade Educator ### Problem 40 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ f(x) = - 4x^2 + 24x - 41 $ Evelyn C. Numerade Educator ### Problem 41 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ g(x) = \frac{1}{2} (x^2 + 4x - 2) $JH J H. Numerade Educator ### Problem 42 In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.$ f(x) = \frac{3}{5} (x^2 + 6x - 5) $ Erik S. Numerade Educator ### Problem 43 In Exercises 43-46, write an equation for the parabola in standard form. JH J H. Numerade Educator ### Problem 44 In Exercises 43-46, write an equation for the parabola in standard form. JH J H. Numerade Educator ### Problem 45 In Exercises 43-46, write an equation for the parabola in standard form. JH J H. Numerade Educator ### Problem 46 In Exercises 43-46, write an equation for the parabola in standard form. JH J H. Numerade Educator ### Problem 47 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( -2, 5 ) $; point:$ ( 0, 9 ) $JH J H. Numerade Educator ### Problem 48 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( 4, -1 ) $; point:$ ( 2, 3 ) $ Suzanne W. Numerade Educator ### Problem 49 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( 1, -2 ) $; point:$ ( -1, 14 ) $JH J H. Numerade Educator ### Problem 50 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( 2, 3 ) $; point:$ ( 0, 2 ) $JH J H. Numerade Educator ### Problem 51 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( 5, 12 ) $; point:$ ( 7, 15 ) $JH J H. Numerade Educator ### Problem 52 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( -2, -2 ) $; point:$ ( -1, 0 ) $JH J H. Numerade Educator ### Problem 53 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ \left(-\frac{1}{4}, \frac{3}{2} \right) $; point:$ ( -2, 0 ) $JH J H. Numerade Educator ### Problem 54 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ \left(\frac{5}{2}, -\frac{3}{4} \right) $; point:$ ( -2, 4 ) $JH J H. Numerade Educator ### Problem 55 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ \left(-\frac{5}{2}, 0 \right) $; point:$ \left(-\frac{7}{2}, -\frac{16}{3} \right) $JH J H. Numerade Educator ### Problem 56 In Exercises 47-56, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex:$ ( 6, 6 ) $; point:$ \left(\frac{61}{10}, \frac{3}{2} \right) $JH J H. Numerade Educator ### Problem 57 In Exercises 57 and 58, determine the x-intercept(s) of the graph visually. Then find the x-intercept(s) algebraically to confirm your results.$ y = x^2 - 4x - 5 $JH J H. Numerade Educator ### Problem 58 In Exercises 57 and 58, determine the x-intercept(s) of the graph visually. Then find the x-intercept(s) algebraically to confirm your results.$ y = 2x^2 + 5x - 3 $JH J H. Numerade Educator ### Problem 59 In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when$ f(x) = 0 $.$ f(x) = x^2 - 4x $JH J H. Numerade Educator ### Problem 60 In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when$ f(x) = 0 $.$ f(x) = -2x^2 + 10x $JH J H. Numerade Educator ### Problem 61 In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when$ f(x) = 0 $.$ f(x) = x^2 - 9x + 18 $JH J H. Numerade Educator ### Problem 62 In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when$ f(x) = 0 $.$ f(x) = x^2 - 8x - 20 $JH J H. Numerade Educator ### Problem 63 In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when$ f(x) = 0 $.$ f(x) = 2x^2 - 7x - 30 $JH J H. Numerade Educator ### Problem 64 In Exercises 59-64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when$ f(x) = 0 $.$ f(x) = \frac{7}{10} (x^2 + 12x -45) $ Jeremy S. Numerade Educator ### Problem 65 In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)$ ( -1, 0 ) $,$ ( 3, 0 ) $JH J H. Numerade Educator ### Problem 66 In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)$ ( -5, 0 ) $,$ ( 5, 0 ) $JH J H. Numerade Educator ### Problem 67 In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)$ ( 0, 0 ) $,$ ( 10, 0 ) $JH J H. Numerade Educator ### Problem 68 In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)$ ( 4, 0 ) $,$ ( 8, 0 ) $JH J H. Numerade Educator ### Problem 69 In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)$ ( -3, 0 ) $,$ \left(-\frac{1}{2}, 0 \right) $JH J H. Numerade Educator ### Problem 70 In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)$ \left(-\frac{5}{2}, 0 \right) $,$ ( 2, 0 ) $JH J H. Numerade Educator ### Problem 71 In Exercises 71-74, find two positive real numbers whose product is a maximum. The sum is$ 110 $. JH J H. Numerade Educator ### Problem 72 In Exercises 71-74, find two positive real numbers whose product is a maximum. The sum is$ S $. JH J H. Numerade Educator ### Problem 73 In Exercises 71-74, find two positive real numbers whose product is a maximum. The sum of the first and twice the second is$ 24 $. JH J H. Numerade Educator ### Problem 74 In Exercises 71-74, find two positive real numbers whose product is a maximum. The sum of the first and three times the second is$ 42 $. JH J H. Numerade Educator ### Problem 75 The path of a diver is given by$ y = - \frac{4}{9} x^2 + \frac{24}{9} + 12 $where$ y $is the height (in feet) and$ x $is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver? JH J H. Numerade Educator ### Problem 76 The height$ y $(in feet) of a punted football is given by$ y = - \frac{16}{2025} x^2 + \frac{9}{5} x + 1.5 $where$ x $is the horizontal distance (in feet) from the point at which the ball is punted. (a) How high is the ball when it is punted? (b) What is the maximum height of the punt? (c) How long is the punt? JH J H. Numerade Educator ### Problem 77 A manufacturer of lighting fixtures has daily production costs of$ C = 800 - 10x + 0.25x^2 $, where$ C $is the total cost (in dollars) and$ x $is the number of units produced. How many fixtures should be produced each day to yield a minimum cost? JH J H. Numerade Educator ### Problem 78 The profit$ P $(in hundreds of dollars) that a company makes depends on the amount$ x $(in hundreds of dollars) the company spends on advertising according to the model$ P = 230 + 20x - 0.5x^2 $. What expenditure for advertising will yield a maximum profit? JH J H. Numerade Educator ### Problem 79 The total revenue$ R $earned (in thousands of dollars) from manufacturing handheld video games is given by$ R(p) = -25p^2 + 1200p $where$ p $is the price per unit (in dollars). (a) Find the revenues when the price per unit is$ \$20$, $\$25 $, and$ \$30$.
(b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain Jeremy S.
The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by $R(p) = - 12 p^2 + 150p$, where $p$ is the price charged per pet (in dollars).