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Homework, Page 269
Chapter Review
Write an equation for the linear function f satisfying the given
conditions. Graph y = f (x).
1. f  3  2 and f  4   9
9   2 
7
m

 1  y   9     x  4 
4   3
7
y
y  9   x  4 or y   x  5





      







x






Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 1
Homework, Page 269
Chapter Review
Find the vertex and axis of the graph of the function. Support
graphically.
5. f  x   2  x  3  5
2
2
f  x   2  x  3  5  y  k  a  x  h 
2
y  a  x  h   k  h  3, k  5  vertex  3,5 
2
axis of symmetry x  3
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Slide 2- 2
Homework, Page 269
Chapter Review
Write an equation for the quadratic function whose graph
contains the given vertex and point.
9. Vertex  2, 3 , point 1,2 
Vertex  2, 3 , point 1, 2   y   3  a  x   2  
y  3  a  x  2   2  3  a 1  2 
2
2
2
5
 5  a 9   a 
9
5
5
2
2
y  3   x  2  or y   x  2   3
9
9
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Slide 2- 3
Homework, Page 269
Chapter Review
Graph the function in a viewing window that shows all of its
extrema and x-intercepts.
2
f
x

x
 3x  40
13  
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Slide 2- 4
Homework, Page 269
Chapter Review
Write the statement as a power function equation. Let k be the
constant of variation
17. The surface area of a sphere varies directly as the
square of the radius.
S  kr
2
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Slide 2- 5
Homework, Page 269
Chapter Review
Write the values of the constants k and a for a function y = k·xa.
Describe the curve that lies in Quadrant I or IV. Determine whether
f is even, odd, or undefined for x < 0. Describe the rest of the curve,
if any. Graph the function to see whether it matches the description.
21. f  x   4 x
1
3
For the function f  x   4 x 3 , k  4 and a  1 . The portion
3
of the curve in Quadrant I is similar in shape to the squareroot
function. The function f is odd, continuing into Quadrant III.
1
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Slide 2- 6
Homework, Page 269
Chapter Review
Divide f (x) by d (x), and write a summary statement in
polynomial form.
25. f  x   2 x  7 x  4 x  5; dx  x  3
3 2 7 4 5
3
2
6 3 3
2 1 1 2


2
2 x  7 x  4 x  5   x  3  2 x  x  1 
x3
3
2
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2
Slide 2- 7
Homework, Page 269
Chapter Review
Use the Remainder Theorem to find the remainder when f (x) is
divided by x – k. Check by using synthetic division.
29. f  x   3x  2 x  x  5; k  2
3
2
f  2   3  2   2  2    2   5  39
3
2
2
3 2 1 5
6 16 34
3 8 17 39
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Slide 2- 8
Homework, Page 269
Chapter Review
Use synthetic division to prove that the number k is an upper
bound for the real zeros of the function f.
33. k  5; f  x   x  5 x  3x  4
3
5
2
1 5 3 4
5 0 15
1 0 3 19
Since the quotient is all nonnegative numbers, 5
is an upper limit of the function.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 9
Homework, Page 269
Chapter Review
Use the Rational Zero Theorem to write a list of all potential
rational zeros. Then determine which, if any, are.
37.
f  x  2x  x  4x  x  6
4
3
2
1 3
f  x   2 x  x  4 x  x  6  Pos : 1, 2, 3, 6,  , 
2 2
2
2 1 4 1 6
4 6 4 6
2 3 2 3 0
3

3 0 3
2
2 0 2 0
4
3
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 10
Homework, Page 269
Chapter Review
Perform the indicated operation and write the result in the form
a + bi.
29
i
41.
i 29  i 28 i  i
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Slide 2- 11
Homework, Page 269
Chapter Review
Match the polynomial function with the graph. Explain your
answer.
45. f  x    x  2 
2
c.
The function and the graph match because the
function has a zero at x = 2 and the graph does
not cross the x-axis at that point. Graph b is
eliminated because it’s y-intercept is too great.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 12
Homework, Page 269
Chapter Review
Find all of the real zeros of the function, finding exact values
whenever possible. Identify each zero as rational or irrational.
State the number of nonreal complex zeros.
49. f  x   x  10 x  23x
4
3
2

f  x   x 4  10 x 3  23 x 2  f  x   x 2 x 2  10 x  23
x
  10  
 10 
2 1
2
 4 1 23

10  100  92

2
10  8 10  2 2
x

 5 2
2
2

Rational zeros : x  0,0;Irrational zeros : x  5  2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 2- 13
Homework, Page 269
Chapter Review
Find all of the zeros and write a linear factorization of the function.
3
2
f
x

2
x

9
x
 2 x  30
53.  
3

2
2
9
2
30
3 18 30  2 x 2  12 x  20  0
2 12 20 0
x
x
  12  
 12 
2  2
2
 4  2  20 

12  144  160
4
12  16 12  4i
3


 3  i  f  x    x    x  3  i  x  3  i 
4
4
2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 14
Homework, Page 269
Chapter Review
Write the function as a product of linear and irreducible quadratic
factors, all with real coefficients.
57.
f  x   x3  x 2  x  2
f  x   x3  x 2  x  2
2
1 1 1 2
2 2 2
1 1 1 0


f  x    x  2 x2  x  1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 15
Homework, Page 269
Chapter Review
Write a polynomial function with real coefficients whose zeros
and their multiplicities include those listed..
61. Degree 3; zeros : 5,  5,3





f  x   x  5 x  5  x  3  x  5  x  3
2
f  x   x  3x  5 x  15
3
2
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Slide 2- 16
Homework, Page 269
Chapter Review
Write a polynomial function with real coefficients whose zeros
and their multiplicities include those listed.
65. Degree 4; zeros : 2  multiplicity 2  ,4  multiplicity 2 


f  x    x  2   x  4   x 2  4 x  4 x 2  8 x  16
2
2

 x  8 x  16 x  4 x  32 x  64 x  4 x  32 x  64
4
3
2
3
2
2
 x 4  4 x3  12 x 2  32 x  64
f  x   x  4 x  12 x  32 x  64
4
3
2
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Slide 2- 17
Homework, Page 269
Chapter Review
Find the asymptotes and intercepts of the function and graph it.
x  x 1
69. f  x   x 2  1
2
Vertical asymptotes : x  1, x  1
Horizontal asymptote : y  1
x  intercept : none
y  intercept : y  1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y
x
Slide 2- 18
Homework, Page 269
Chapter Review
Find the intercepts and analyze and graph the rational function.
x  x  2x  5
f  x 
x2
3
73.
2
Vertical asymptote : x  2
No horizontal asymptotes
x  intercept : x  2.552
5
y  intercept : y 
2
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y
x
Slide 2- 19
Homework, Page 269
Chapter Review
Solve the equation algebraically and support graphically.
3
2
2
x

3
x
 17 x  30  0
77.
3
2 3 17 30
6 27 30  2 x 2  9 x  10  0
2 9 10
0
 2 x  5 x  2   0   x  3 2 x  5  x  2   0
x  2.5 2.5  x  2 2  x  3 x  3




 , 2.5  2,3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 20
Homework, Page 269
Chapter Review
Solve the equation algebraically and support graphically.
3
2
2
x

3
x
 17 x  30  0
77.
 , 2.5  2,3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 21
Homework, Page 269
Chapter Review
Solve the equation algebraically and support graphically.
2
x

1
x

3

0




81.
2
1
x  3 x  3 3  x 
2

0

1
x
2
0
1
x
2

1

 , 
2

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Slide 2- 22
Homework, Page 269
Chapter Review
85. Villareal Paper Co. has contracted to
manufacture a box with no top that is to be
made by removing squares of width x from
the corners of a 30-in by 70-in piece of
cardboard.
a. Find an equation that models the volume of
the box.
b. Determine x so that the box has volume 5800
in3.
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Slide 2- 23
Homework, Page 269
Chapter Review
85. a. Find an equation that models the volume
of the box.
V  lwh   70  2 x  30  2 x  x
b. Determine x so that the box has volume 5800
in3.
x  4.567 in or x  8.627 in
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Homework, Page 269
Chapter Review
89. The table shows the spending at NIH for
several years. Let x = 0 represent 1990, etc.
Year
Amt (109)
Year
Amt (109)
1993
10.3
1998
13.6
1994
11.0
1999
15.6
1995
11.3
2000
17.9
1996
11.9
2001
20.5
1997
12.7
2002
23.6
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Slide 2- 25
Homework, Page 269
Chapter Review
89. a. Find a linear regression model, and graph
it together with a scatter plot of the data.
y  1.401x  4.331
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Slide 2- 26
Homework, Page 269
Chapter Review
89. b. Find a quadratic regression model and
graph it with a scatter plot of the data.
y  0.1875 x  1.411x  13.331
2
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Slide 2- 27
Homework, Page 269
Chapter Review
89. c. Use the linear and quadratic models to
estimate when the amount of spending will
exceed $30-billion annually.
By the linear model, the spending level reaches
$30-billion in the 19th year (2009). By the
quadratic model, it reaches the $30-billion in
the 14th year (2004).
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Slide 2- 28
Homework, Page 269
Chapter Review
93. Suppose x ounces of distilled water are added
to 50 oz of pure acid.
a. Express the concentration C (x) of the new
mixture as a function of x.
b. Use a graph to determine how much distilled
water should be added to the pure acid to
produce a new solution that is less than 60%
acid.
c. Solve (b) algebraically.
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Slide 2- 29
Homework, Page 269
Chapter Review
93. a. Express the concentration C (x) of the new
mixture as a function of x.
50
C  x 
50  x
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Slide 2- 30
Homework, Page 269
Chapter Review
93. b. Use a graph to determine how much
distilled water should be added to the pure
acid to produce a new solution that is less
than 60% acid.
x  33.3 oz.
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Slide 2- 31
Homework, Page 269
Chapter Review
93. c. Solve (b) algebraically.
50
 0.6  50  30  0.6 x  0.6 x  20  x  33.3 oz.
50  x
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Slide 2- 32
Chapter Test
1. Write an equation for the linear function f satisfying the given condition:
f (3)  2 and f (4)  9.
2. Write an equation for the quadratic function whose graph contains the
vertex (  2, 3) and the point (1,2).
3. Write the statement as a power function equation. Let k be the constant
of variation. The surface area S of a sphere varies directly as the square of
the radius r.
4. Divide f ( x) by d ( x), and write a summary statement in polynomial form:
f ( x)  2 x  7 x  4 x  5; d ( x)  x  3
5. Use the Rational Zeros Theorem to write a list of all potential rational
zeros. Then determine which ones, if any, are zeros.
f ( x)  2 x  x  4 x  x  6
3
4
2
3
2
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Slide 2- 33
Chapter Test
6. Find all zeros of the function. f ( x)  x  10 x  23x
7. Find all zeros and write a linear factorization of the function.
f ( x)  5 x  24 x  x  12
4
3
3
2
2
8. Find the asymptotes and intercepts of the function.
x  x 1
f ( x) 
x 1
9. Solve the equation or inequality algebraically.
12
2 x   11
x
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 34
Chapter Test
10. Larry uses a slingshot to launch a rock straight up from a point
6 ft above level ground with an initial velocity of 170 ft/sec.
(a) Find an equation that models the height of the rock t
seconds after it is launched.
(b) What is the maximum height of the rock?
(c) When will it reach that height?
(d) When will the rock hit the ground?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 35
Chapter Test Solutions
1. Write an equation for the linear function f satisfying the given condition:
f (3)  2 and f (4)  9. y   x  5
2. Write an equation for the quadratic function whose graph contains the
vertex (  2, 3) and the point (1,2). y  5 / 9( x  2)  3
2
3. Write the statement as a power function equation. Let k be the constant
of variation. The surface area S of a sphere varies directly as the square of
the radius r. s  kr
4. Divide f ( x) by d ( x), and write a summary statement in polynomial form:
2
f ( x)  2 x  7 x  4 x  5; d ( x)  x  3 2x  x  1 
x 3
5. Use the Rational Zeros Theorem to write a list of all potential rational
zeros. Then determine which ones, if any, are zeros.
f ( x)  2 x  x  4 x  x  6  1, 2, 3, 6  1/ 2, 3 / 2;  3 / 2 and 2
2
3
4
2
3
2
2
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Slide 2- 36
Chapter Test Solutions
6. Find all zeros of the function. f ( x)  x  10 x  23x 0,5  2
7. Find all zeros and write a linear factorization of the function.
4
3
2
f ( x)  5 x  24 x  x  12
4 / 5, 2  7
8. Find the asymptotes and intercepts of the function.
x  x 1
f ( x) 
y -intercept (0,1), x-intercept none
x 1
Vertical Asymptote x  1, x  1, Horizontal Asymptote y  1
9. Solve the equation or inequality algebraically.
12
2 x   11 x  3 / 2 or x  4
x
3
2
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 37
Chapter Test Solutions
10. Larry uses a slingshot to launch a rock straight up from a point
6 ft above level ground with an initial velocity of 170 ft/sec.
(a) Find an equation that models the height of the rock t
seconds after it is launched. h  -16t  170t  6
2
(b) What is the maximum height of the rock? 457.5625ft
(c) When will it reach that height? 5.3125 sec
(d) When will the rock hit the ground? 10.66 sec
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 38