Stewart Precalc 6e Section 3.1: Quadratic Functions and Models

1 Concept - Standard form · Level 1

To put the quadratic function $f(x) = a x^2 + b x + c$ in standard form, we complete the ____.

2 Concept - Vertex and orientation · Level 1

The quadratic function $f(x) = a(x - h)^2 + k$ is in standard form. (a) The graph of $f$ is a parabola with vertex (____, ____). (b) If $a > 0$, the graph of $f$ opens ____. In this case $f(h) = k$ is the ____ value of $f$. (c) If $a < 0$, the graph of $f$ opens ____. In this case $f(h) = k$ is the ____ value of $f$.

3 Concept - Vertex identification · Level 1

The graph of $f(x) = 2(x - 3)^2 + 5$ is a parabola that opens ____, with its vertex at (____, ____), and $f(3) = $ ____ is the (minimum/maximum) ____ value of $f$.

4 Concept - Vertex identification · Level 1

The graph of $f(x) = -2(x - 3)^2 + 5$ is a parabola that opens ____, with its vertex at (____, ____), and $f(3) = $ ____ is the (minimum/maximum) ____ value of $f$.

5 Skills - Graph reading · Level 2

The graph of a quadratic function $f$ is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of $f$. (c) Find the domain and range of $f$. $f(x) = -x^2 + 6 x - 5$

6 Skills - Graph reading · Level 2

The graph of a quadratic function $f$ is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of $f$. (c) Find the domain and range of $f$. $f(x) = -\dfrac{1}{2} x^2 - 2 x + 6$

7 Skills - Graph reading · Level 2

The graph of a quadratic function $f$ is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of $f$. (c) Find the domain and range of $f$. $f(x) = 2 x^2 - 4 x - 1$

8 Skills - Graph reading · Level 2

The graph of a quadratic function $f$ is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of $f$. (c) Find the domain and range of $f$. $f(x) = 3 x^2 + 6 x - 1$

9 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = x^2 - 6 x$

10 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = x^2 + 8 x$

11 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = 2 x^2 + 6 x$

12 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = -x^2 + 10 x$

13 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = x^2 + 4 x + 3$

14 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = x^2 - 2 x + 2$

15 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = -x^2 + 6 x + 4$

16 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = -x^2 - 4 x + 4$

17 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = 2 x^2 + 4 x + 3$

18 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = -3 x^2 + 6 x - 2$

19 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = 2 x^2 + 4 x + 5$

20 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = 2 x^2 + x - 6$

21 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = -4 x^2 - 16 x + 3$

22 Skills - Standard form and graphing · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its $x$- and $y$-intercept(s). (c) Sketch its graph. $f(x) = 6 x^2 + 12 x - 5$

23 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $f(x) = x^2 + 2 x - 1$

24 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $f(x) = x^2 - 8 x + 8$

25 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $f(x) = 3 x^2 - 6 x + 1$

26 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $f(x) = 5 x^2 + 30 x + 4$

27 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $f(x) = -x^2 - 3 x + 3$

28 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $f(x) = 1 - 6 x - x^2$

29 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $g(x) = 3 x^2 - 12 x + 13$

30 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $g(x) = 2 x^2 + 8 x + 11$

31 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $h(x) = 1 - x - x^2$

32 Skills - Max/min value · Level 2

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $h(x) = 3 - 4 x - 4 x^2$

33 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(x) = x^2 + x + 1$

34 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(x) = 1 + 3 x - x^2$

35 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(t) = 100 - 49 t - 7 t^2$

36 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(t) = 10 t^2 + 40 t + 113$

37 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(s) = s^2 - 1.2 s + 16$

38 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $g(x) = 100 x^2 - 1500 x$

39 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $h(x) = \dfrac{1}{2} x^2 + 2 x - 6$

40 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(x) = -\dfrac{x^2}{3} + 2 x + 7$

41 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $f(x) = 3 - x - \dfrac{1}{2} x^2$

42 Skills - Max/min value · Level 2

Find the maximum or minimum value of the function. $g(x) = 2 x(x - 4) + 7$

43 Skills - Function from vertex · Level 3

Find a function whose graph is a parabola with vertex $(1, -2)$ and that passes through the point $(4, 16)$.

44 Skills - Function from vertex · Level 3

Find a function whose graph is a parabola with vertex $(3, 4)$ and that passes through the point $(1, -8)$.

45 Skills - Domain and range · Level 2

Find the domain and range of the function. $f(x) = -x^2 + 4 x - 3$

46 Skills - Domain and range · Level 2

Find the domain and range of the function. $f(x) = x^2 - 2 x - 3$

47 Skills - Domain and range · Level 2

Find the domain and range of the function. $f(x) = 2 x^2 + 6 x - 7$

48 Skills - Domain and range · Level 2

Find the domain and range of the function. $f(x) = -3 x^2 + 6 x + 4$

49 Skills - Graphing device · Level 3

A quadratic function is given. (a) Use a graphing device to find the maximum or minimum value of the quadratic function $f$, correct to two decimal places. (b) Find the exact maximum or minimum value of $f$, and compare it with your answer to part (a). $f(x) = x^2 + 1.79 x - 3.21$

50 Skills - Graphing device · Level 3

A quadratic function is given. (a) Use a graphing device to find the maximum or minimum value of the quadratic function $f$, correct to two decimal places. (b) Find the exact maximum or minimum value of $f$, and compare it with your answer to part (a). $f(x) = 1 + x - \sqrt{2} x^2$

51 Skills - Local extrema from graph · Level 2

Find all local maximum and minimum values of the function whose graph is shown.

52 Skills - Local extrema from graph · Level 2

Find all local maximum and minimum values of the function whose graph is shown.

53 Skills - Local extrema from graph · Level 2

Find all local maximum and minimum values of the function whose graph is shown.

54 Skills - Local extrema from graph · Level 2

Find all local maximum and minimum values of the function whose graph is shown.

55 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $f(x) = x^3 - x$

56 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $f(x) = 3 + x + x^2 - x^3$

57 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $g(x) = x^4 - 2 x^3 - 11 x^2$

58 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $g(x) = x^5 - 8 x^3 + 20 x$

59 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $U(x) = x \sqrt{6 - x}$

60 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $U(x) = x \sqrt{x - x^2}$

61 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $V(x) = \dfrac{1 - x^2}{x^3}$

62 Skills - Local extrema · Level 3

Find the local maximum and minimum values of the function and the value of $x$ at which each occurs. State each answer correct to two decimal places. $V(x) = \dfrac{1}{x^2 + x + 1}$

63 Application - Projectile motion · Level 2

Height of a Ball. If a ball is thrown directly upward with a velocity of 40 ft/s, its height (in feet) after $t$ seconds is given by $y = 40 t - 16 t^2$. What is the maximum height attained by the ball?

64 Application - Projectile motion · Level 3

Path of a Ball. A ball is thrown across a playing field from a height of 5 ft above the ground at an angle of $45^{\circ}$ to the horizontal at a speed of 20 ft/s. It can be deduced from physical principles that the path of the ball is modeled by the function $y = -\dfrac{32}{20^2} x^2 + x + 5$ where $x$ is the distance in feet that the ball has traveled horizontally. (a) Find the maximum height attained by the ball. (b) Find the horizontal distance the ball has traveled when it hits the ground.

65 Application - Revenue maximization · Level 2

Revenue. A manufacturer finds that the revenue generated by selling $x$ units of a certain commodity is given by the function $R(x) = 80 x - 0.4 x^2$, where the revenue $R(x)$ is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

66 Application - Profit maximization · Level 2

Sales. A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells $x$ cans of soda pop in one day, his profit (in dollars) is given by $P(x) = -0.001 x^2 + 3 x - 1800$. What is his maximum profit per day, and how many cans must he sell for maximum profit?

67 Application - Advertising effectiveness · Level 2

Advertising. The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness $E$ is measured on a scale of 0 to 10, then $E(n) = \dfrac{2}{3} n - \dfrac{1}{90} n^2$ where $n$ is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?

68 Application - Pharmaceuticals · Level 2

Pharmaceuticals. When a certain drug is taken orally, the concentration of the drug in the patient's bloodstream after $t$ minutes is given by $C(t) = 0.06 t - 0.0002 t^2$, where $0 \leq t \leq 240$ and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?

69 Application - Agriculture optimization · Level 2

Agriculture. The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If $n$ trees are planted on an acre of land, then each tree produces $900 - 9 n$ apples. So the number of apples produced per acre is $A(n) = n(900 - 9 n)$. How many trees should be planted per acre to obtain the maximum yield of apples?

70 Application - Agriculture optimization · Level 3

Agriculture. At a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $A(n) = (700 + n)(10 - 0.01 n)$ where $n$ is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

71 Application - Alternative method · Level 2

Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 220-221. Problem 21.

72 Application - Alternative method · Level 2

Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 220-221. Problem 22.

73 Application - Alternative method · Level 2

Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 220-221. Problem 25.

74 Application - Alternative method · Level 2

Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 220-221. Problem 24.

75 Application - Geometry optimization · Level 3

Fencing a Horse Corral. Carol has 2400 ft of fencing to fence in a rectangular horse corral. (a) Find a function that models the area of the corral in terms of the width $x$ of the corral. (b) Find the dimensions of the rectangle that maximize the area of the corral.

76 Application - Geometry optimization · Level 3

Making a Rain Gutter. A rain gutter is formed by bending up the sides of a 30-inch-wide rectangular metal sheet as shown in the figure. (a) Find a function that models the cross-sectional area of the gutter in terms of $x$. (b) Find the value of $x$ that maximizes the cross-sectional area of the gutter. (c) What is the maximum cross-sectional area for the gutter?

77 Application - Revenue maximization · Level 3

Stadium Revenue. A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at \$10, the average attendance at recent games has been 27,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000. (a) Find a function that models the revenue in terms of ticket price. (b) Find the price that maximizes revenue from ticket sales. (c) What ticket price is so high that no revenue is generated?

78 Application - Profit maximization · Level 3

Maximizing Profit. A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost \$6, and the society sells an average of 20 per week at a price of \$10 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

79 Discovery - Vertex and intercepts · Level 3

Vertex and $x$-Intercepts. We know that the graph of the quadratic function $f(x) = (x - m)(x - n)$ is a parabola. Sketch a rough graph of what such a parabola would look like. What are the $x$-intercepts of the graph of $f$? Can you tell from your graph the $x$-coordinate of the vertex in terms of $m$ and $n$? (Use the symmetry of the parabola.) Confirm your answer by expanding and using the formulas of this section.

80 Discovery - Substitution method · Level 3

Maximum of a Fourth-Degree Polynomial. Find the maximum value of the function $f(x) = 3 + 4 x^2 - x^4$. [Hint: Let $t = x^2$.]

81 Example - Standard Form of a Quadratic Function · Level 2

Let $f(x) = 2 x^2 - 12 x + 23$.

(a) Express $f$ in standard form.

Answer for 81(a)

(b) Sketch the graph of $f$.

Answer for 81(b)

Enter your answer directly below each part above.

82 Example - Minimum Value of a Quadratic Function · Level 2

Consider the quadratic function $f(x) = 5 x^2 - 30 x + 49$.

(a) Express $f$ in standard form.

Answer for 82(a)

(b) Sketch the graph of $f$.

Answer for 82(b)

(c) Find the minimum value of $f$.

Answer for 82(c)

Enter your answer directly below each part above.

83 Example - Maximum Value of a Quadratic Function · Level 2

Consider the quadratic function $f(x) = -x^2 + x + 2$.

(a) Express $f$ in standard form.

Answer for 83(a)

(b) Sketch the graph of $f$.

Answer for 83(b)

(c) Find the maximum value of $f$.

Answer for 83(c)

Enter your answer directly below each part above.

84 Example - Finding Maximum and Minimum Values of Quadratic Functions · Level 2

Find the maximum or minimum value of each quadratic function.

(a) $f(x) = x^2 + 4 x$

Answer for 84(a)

(b) $g(x) = -2 x^2 + 4 x - 5$

Answer for 84(b)

Enter your answer directly below each part above.

85 Example - Maximum gas mileage · Level 2

Most cars get their best gas mileage when traveling at a relatively modest speed. The gas mileage $M$ for a certain new car is modeled by the function $M(s) = -\dfrac{1}{28} s^2 + 3 s - 31, 15 \leq s \leq 70$ where $s$ is the speed in mi/h and $M$ is measured in mi/gal. What is the car's best gas mileage, and at what speed is it attained?

86 Example - Maximizing ticket revenue · Level 3

A hockey team plays in an arena that has a seating capacity of 15,000 spectators. With the ticket price set at \$14, average attendance at recent games has been 9500. A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 1000. (a) Find a function that models the revenue in terms of ticket price. (b) Find the price that maximizes revenue from ticket sales. (c) What ticket price is so high that no one attends and so no revenue is generated?