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Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 3

Polynomial and Rational Functions - all with Video Answers

Educators

Section 1

Quadratic Functions and Models

00:28

Problem 1

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

Daniel Nolan
Daniel Nolan
Numerade Educator
01:43

Problem 2

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 $f$.

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
02:49

Problem 3

The graph of $f(x)=3(x-2)^{2}-6$ is a parabola that opens ______ , with its vertex at ( ____ , ____ ) and $f(2)=$ ________ is the (minimum/maximum) ______ value of $f$.

Daniel Nolan
Daniel Nolan
Numerade Educator
01:13

Problem 4

The graph of $f(x)=-3(x-2)^{2}-6$ is a parabola that that opens ______ , with its vertex at ( ____ , ____ ) and $f(2)=$ ________ is the (minimum/maximum) ______ value of $f$.

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
07:25

Problem 5

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

Daniel Nolan
Daniel Nolan
Numerade Educator
04:26

Problem 6

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

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
06:00

Problem 7

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

Daniel Nolan
Daniel Nolan
Numerade Educator
04:08

Problem 8

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

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
05:00

Problem 9

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=x^{2}-2 x+3$

Daniel Nolan
Daniel Nolan
Numerade Educator
04:26

Problem 10

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=x^{2}+4 x-1$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
05:00

Problem 11

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=x^{2}-6 x$

Daniel Nolan
Daniel Nolan
Numerade Educator
04:15

Problem 12

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=x^{2}+8 x$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
05:10

Problem 13

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=3 x^{2}+6 x$

Daniel Nolan
Daniel Nolan
Numerade Educator
04:16

Problem 14

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=-x^{2}+10 x$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
04:51

Problem 15

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=x^{2}+4 x+3$

Daniel Nolan
Daniel Nolan
Numerade Educator
03:32

Problem 16

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=x^{2}-2 x+2$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
06:02

Problem 17

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=-x^{2}+6 x+4$

Daniel Nolan
Daniel Nolan
Numerade Educator
04:29

Problem 18

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=-x^{2}-4 x+4$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
05:17

Problem 19

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=2 x^{2}+4 x+3$

Daniel Nolan
Daniel Nolan
Numerade Educator
05:37

Problem 20

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=-3 x^{2}+6 x-2$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
05:00

Problem 21

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=2 x^{2}-20 x+57$

Daniel Nolan
Daniel Nolan
Numerade Educator
04:50

Problem 22

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=2 x^{2}+12 x+10$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
09:53

Problem 23

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=-4 x^{2}-12 x+1$

Daniel Nolan
Daniel Nolan
Numerade Educator
06:45

Problem 24

Graphing Quadratic Functions A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Find the vertex and $x$- and $y$-intercepts of $f$. (c) Sketch a graph of $f .$ (d) Find the domain and range of $f.$
$f(x)=3 x^{2}+2 x-2$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
04:54

Problem 25

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

Daniel Nolan
Daniel Nolan
Numerade Educator
03:30

Problem 26

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

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
04:25

Problem 27

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

Daniel Nolan
Daniel Nolan
Numerade Educator
02:56

Problem 28

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

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
05:11

Problem 29

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

Daniel Nolan
Daniel Nolan
Numerade Educator
02:48

Problem 30

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

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
04:04

Problem 31

Maximum and Minimum Values A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Sketch a graph of $f$ . (c) Find the maximum or minimum value of $f.$
$g(x)=3 x^{2}-12 x+13$

Daniel Nolan
Daniel Nolan
Numerade Educator
02:33

Problem 32

Maximum and Minimum Values A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Sketch a graph of $f$ . (c) Find the maximum or minimum value of $f.$
$g(x)=2 x^{2}+8 x+11$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
04:32

Problem 33

Maximum and Minimum Values A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Sketch a graph of $f$ . (c) Find the maximum or minimum value of $f.$
$h(x)=1-x-x^{2}$

Daniel Nolan
Daniel Nolan
Numerade Educator
02:27

Problem 34

Maximum and Minimum Values A quadratic function $f$ is given. (a) Express $f$ in standard form. (b) Sketch a graph of $f$ . (c) Find the maximum or minimum value of $f.$
$h(x)=3-4 x-4 x^{2}$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
01:39

Problem 35

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(x)=2 x^{2}+4 x-1$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:43

Problem 36

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(x)=3-4 x-x^{2}$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
01:55

Problem 37

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(t)=-3+80 t-20 t^{2}$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:20

Problem 38

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(x)=6 x^{2}-24 x-100$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
02:30

Problem 39

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(s)=s^{2}-1.2 s+16$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:44

Problem 40

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$g(x)=100 x^{2}-1500 x$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
01:26

Problem 41

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$h(x)=\frac{1}{2} x^{2}+2 x-6$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:17

Problem 42

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(x)=-\frac{x^{2}}{3}+2 x+7$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
01:36

Problem 43

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$f(x)=3-x-\frac{1}{2} x^{2}$

Daniel Nolan
Daniel Nolan
Numerade Educator
02:12

Problem 44

Formula for Maximum and Minimum Values. Find the maximum or minimum value of the function.
$g(x)=2 x(x-4)+7$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
03:35

Problem 45

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

Daniel Nolan
Daniel Nolan
Numerade Educator
03:32

Problem 46

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

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
03:11

Problem 47

Finding Quadratic Functions Find a function $f$ whose graph is a parabola with the given vertex and that passes through the given point.
Vertex $(2,-3) ; \quad$ point $(3,1)$

Daniel Nolan
Daniel Nolan
Numerade Educator
02:10

Problem 48

Finding Quadratic Functions Find a function $f$ whose graph is a parabola with the given vertex and that passes through the given point.
Vertex $(-1,5) ; \quad$ point $(-3,-7)$

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
03:10

Problem 49

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

Daniel Nolan
Daniel Nolan
Numerade Educator
02:52

Problem 50

Minimum of a Sixth-Degree Polynomial Find the minimum value of the function
$$f(x)=2+16 x^{3}+4 x^{6}$$
$[$Hint$:$ Let $t=x^{3}$.]

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
01:59

Problem 51

Height of a Ball If a ball is thrown directly upward with a velocity of 40 $\mathrm{ft} / \mathrm{s}$ , 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?

Daniel Nolan
Daniel Nolan
Numerade Educator
06:54

Problem 52

Path of a Ball A ball is thrown across a playing field from a height of 5 $\mathrm{ft}$ above the ground at an angle of $45^{\circ}$ to the horizontal at a speed of 20 $\mathrm{ft} / \mathrm{s}$ . It can be deduced from physical principles that the path of the ball is modeled by the function
$$y=-\frac{32}{(20)^{2}} x^{2}+x+5$$
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.

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
02:04

Problem 53

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?

Daniel Nolan
Daniel Nolan
Numerade Educator
02:57

Problem 54

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?

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
02:09

Problem 55

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
$$E(n)=\frac{2}{3} n-\frac{1}{90} n^{2}$$
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?

Daniel Nolan
Daniel Nolan
Numerade Educator
03:18

Problem 56

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?

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
01:34

Problem 57

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?

Daniel Nolan
Daniel Nolan
Numerade Educator
01:50

Problem 58

Agriculture At a certain vineyard it is found that each grape vine produces about 10 lb 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.

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
03:26

Problem 59

Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages $237-244$ .
Problem 21

Cinsy Krehbiel
Cinsy Krehbiel
Numerade Educator
05:00

Problem 60

Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages $237-244$ .
Problem 22

Cinsy Krehbiel
Cinsy Krehbiel
Numerade Educator
09:35

Problem 61

Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages $237-244$ .
Problem 25

Cinsy Krehbiel
Cinsy Krehbiel
Numerade Educator
04:50

Problem 62

Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages $237-244$ .
Problem 24

Cinsy Krehbiel
Cinsy Krehbiel
Numerade Educator
04:14

Problem 63

Fencing a Horse Corral Carol has 2400 $\mathrm{ft}$ 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.

Daniel Nolan
Daniel Nolan
Numerade Educator
03:12

Problem 64

Making a Rain Gutter A rain gutter is formed by bending up the sides of a $30-$ in.-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?

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
06:45

Problem 65

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?

Daniel Nolan
Daniel Nolan
Numerade Educator
05:58

Problem 66

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 will lose 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?

Vishnu Rathnam
Vishnu Rathnam
Numerade Educator
04:00

Problem 67

DISCOVER: 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.

Daniel Nolan
Daniel Nolan
Numerade Educator