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College Algebra 7th

James Stewart, Lothar Redlin, Saleem Watson

Chapter 3

Polynomial and Rational Functions

Educators

AG

Problem 1

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

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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 $f$

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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 $f$

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Problem 4

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 $f$.

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Problem 5

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 $f$
$$f(x)=-x^{2}+6 x-5$$

AG
Ankit G.
Numerade Educator

Problem 6

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 $f$
$$f(x)=-\frac{1}{2} x^{2}-2 x+6$$

AG
Ankit G.
Numerade Educator

Problem 7

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 $f$
$$f(x)=2 x^{2}-4 x-1$$

AG
Ankit G.
Numerade Educator

Problem 8

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 $f$
$$f(x)=3 x^{2}-6 x-1$$

AG
Ankit G.
Numerade Educator

Problem 9

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 $f$
$$f(x)=x^{2}-2 x+3$$

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Problem 10

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 $f$
$$f(x)=x^{2}+4 x-1$$

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Problem 11

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 $f$
$$f(x)=x^{2}-6 x$$

AG
Ankit G.
Numerade Educator

Problem 12

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 $f$
$$f(x)=x^{2}+8 x$$

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Problem 13

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 $f$
$$f(x)=3 x^{2}+6 x$$

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Problem 14

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 $f$
$$f(x)=-x^{2}+10 x$$

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Problem 15

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 $f$
$$f(x)=x^{2}+4 x+3$$

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Problem 16

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 $f$
$$f(x)=x^{2}-2 x+2$$

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Problem 17

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 $f$
$$f(x)=-x^{2}+6 x+4$$

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Problem 18

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 $f$
$$f(x)=-x^{2}-4 x+4$$

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Problem 19

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 $f$
$$f(x)=2 x^{2}+4 x+3$$

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Problem 20

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 $f$
$$f(x)=-3 x^{2}+6 x-2$$

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Problem 21

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 $f$
$$f(x)=2 x^{2}-20 x+57$$

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Problem 22

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 $f$
$$f(x)=2 x^{2}+12 x+10$$

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Problem 23

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 $f$
$$f(x)=-4 x^{2}-12 x+1$$

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Problem 24

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 $f$
$$f(x)=3 x^{2}+2 x-2$$

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Problem 25

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 $f$
$$f(x)=x^{2}+2 x-1$$

AG
Ankit G.
Numerade Educator

Problem 26

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 $f$
$$f(x)=x^{2}-8 x+8$$

AG
Ankit G.
Numerade Educator

Problem 27

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 $f$
$$f(x)=3 x^{2}-6 x+1$$

AG
Ankit G.
Numerade Educator

Problem 28

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 $f$
$$f(x)=5 x^{2}+30 x+4$$

AG
Ankit G.
Numerade Educator

Problem 29

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 $f$
$$f(x)=-x^{2}-3 x+3$$

AG
Ankit G.
Numerade Educator

Problem 30

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 $f$
$$f(x)=1-6 x-x^{2}$$

AG
Ankit G.
Numerade Educator

Problem 31

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 $f$
$$g(x)=3 x^{2}-12 x+13$$

AG
Ankit G.
Numerade Educator

Problem 32

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 $f$
$$g(x)=2 x^{2}+8 x+11$$

AG
Ankit G.
Numerade Educator

Problem 33

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 $f$
$$h(x)=1-x-x^{2}$$

AG
Ankit G.
Numerade Educator

Problem 34

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 $f$
$$h(x)=3-4 x-4 x^{2}$$

AG
Ankit G.
Numerade Educator

Problem 35

Find the maximum or minimum value of the function.
$$f(x)=2 x^{2}+4 x-1$$

AG
Ankit G.
Numerade Educator

Problem 36

Find the maximum or minimum value of the function.
$$f(x)=3-4 x-x^{2}$$

AG
Ankit G.
Numerade Educator

Problem 37

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

AG
Ankit G.
Numerade Educator

Problem 38

Find the maximum or minimum value of the function.
$$f(x)=6 x^{2}-24 x-100$$

AG
Ankit G.
Numerade Educator

Problem 39

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

AG
Ankit G.
Numerade Educator

Problem 40

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

AG
Ankit G.
Numerade Educator

Problem 41

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

AG
Ankit G.
Numerade Educator

Problem 42

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

AG
Ankit G.
Numerade Educator

Problem 43

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

AG
Ankit G.
Numerade Educator

Problem 44

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

AG
Ankit G.
Numerade Educator

Problem 45

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).
$$f(x)=x^{2}+1.79 x-3.21$$

AG
Ankit G.
Numerade Educator

Problem 46

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).
$$f(x)=1+x-\sqrt{2} x^{2}$$

AG
Ankit G.
Numerade Educator

Problem 47

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

AG
Ankit G.
Numerade Educator

Problem 48

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

AG
Ankit G.
Numerade Educator

Problem 49

Find the maximum value of the function
$$f(x)=3+4 x^{2}-x^{4}$$

AG
Ankit G.
Numerade Educator

Problem 50

Minimum of a Sixth-Degree Polynomial Find the minimum value of the function
$$f(x)=2+16 x^{3}+4 x^{6}$$

AG
Ankit G.
Numerade Educator

Problem 51

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?

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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.

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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?

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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?

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Problem 55

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?

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Problem 56

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 $\mathrm{mg} / \mathrm{L}$. When is the maximum serum concentration reached, and what is that maximum concentration?

AG
Ankit G.
Numerade Educator

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?

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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.

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Problem 59

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

AG
Ankit G.
Numerade Educator

Problem 60

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

Sheryl E.
Numerade Educator

Problem 61

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

AG
Ankit G.
Numerade Educator

Problem 62

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

AG
Ankit G.
Numerade Educator

Problem 63

Fencing a Horse Corral Carol has $2400 \mathrm{ft}$ of fencing to fence in a rectang 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.

AG
Ankit G.
Numerade Educator

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?

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Problem 65

Stadium Revenue $\Leftrightarrow$ 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?

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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 810 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?

AG
Ankit G.
Numerade Educator

Problem 67

Discover: Vertex and $x$ -Intercepts we know that the graph of the quadratic function If $f(x)=(x-m)(x-n)$ is a parabola. Sketch a rough graph of what such a parabola would look like. $$ \begin{array}{ll}f(x)=(x-m)(x-n) \text { is araph of }(x-m)(x-h) \text { and and and and and and and and and and } \\ \text { the paration }\end{array} $$ 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.

AG
Ankit G.
Numerade Educator