homework, page 269 chapter review

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Homework, Page 269Chapter Review

Write an equation for the linear function f satisfying the given conditions. Graph y = f (x).

1. 3 2 and 4 9f f

9 2 7

1 9 44 3 7

9 4 or 5

m y x

y x y x

x

y


Homework, 69Chapter Review

Find the vertex and axis of the graph of the function. Support graphically.

5. 22 3 5f x x

2 2

2

2 3 5

3, 5 vertex 3,5

axis of symmetry 3

f x x y k a x h

y a x h k h k

x



Write an equation for the quadratic function whose graph contains the given vertex and point.

9. Vertex 2, 3 , point 1,2

2

2 2

2 2

Vertex 2, 3 , point 1,2 3 2

53 2 2 3 1 2 5 9

9

5 53 2 or 2 3

9 9

y a x

y a x a a a

y x y x



Graph the function in a viewing window that shows all of its extrema and x-intercepts.

13 2 3 40f x x x



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.

2S kr



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. 1

34f x x

13 1For the function 4 , 4 and . The portion3

of the curve in Quadrant I is similar in shape to the squareroot

function. The function is odd, continuing into Quadrant III.

f x x k a

f



Divide f (x) by d (x), and write a summary statement in polynomial form.

25. 3 22 7 4 5; 3f x x x x dx x

3 2 2

3 2 7 4 5

6 3 3

2 1 1 2

22 7 4 5 3 2 1

3x x x x x x

x



Use the Remainder Theorem to find the remainder when f (x) is divided by x – k. Check by using synthetic division.

29. 3 23 2 5; 2f x x x x k

3 22 3 2 2 2 2 5 39

2 3 2 1 5

6 16 34

3 8 17 39

f



Use synthetic division to prove that the number k is an upper bound for the real zeros of the function f.

33.

Since the quotient is all nonnegative numbers, 5 is an upper limit of the function.

3 25; 5 3 4k f x x x x

5 1 5 3 4

5 0 15

1 0 3 19



Use the Rational Zero Theorem to write a list of all potential rational zeros. Then determine which, if any, are.

37. 4 3 22 4 6f x x x x x

4 3 2 1 32 4 6 : 1, 2, 3, 6, ,

2 22 2 1 4 1 6

4 6 4 6

2 3 2 3 0

33 0 3

2

2 0 2 0

f x x x x x Pos



Perform the indicated operation and write the result in the form a + bi.

41. 29i

29 28i i i i



Match the polynomial function with the graph. Explain your answer.

45. 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.

22f x x



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. 4 3 210 23f x x x x

4 3 2 2 2

2

10 23 10 23

10 10 4 1 23 10 100 92

2 1 2

10 8 10 2 25 2

2 2

Rational zeros : 0,0 ;Irrational zeros : 5 2

f x x x x f x x x x

x

x

x x



Find all of the zeros and write a linear factorization of the function.

53. 3 22 9 2 30f x x x x

2

2

32 9 2 30

2

3 18 30 2 12 20 0

2 12 20 0

12 12 4 2 20 12 144 160

2 2 4

12 16 12 4 33 3 3

4 4 2

x x

x

ix i f x x x i x i



Write the function as a product of linear and irreducible quadratic factors, all with real coefficients.

57. 3 2 2f x x x x

3 2

2

2

2 1 1 1 2

2 2 2

1 1 1 0

2 1

f x x x x

f x x x x



Write a polynomial function with real coefficients whose zeros and their multiplicities include those listed..

61. Degree 3; zeros : 5, 5,3

2

3 2

5 5 3 5 3

3 5 15

f x x x x x x

f x x x x



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

2 2 2 2

4 3 2 3 2 2

4 3 2

4 3 2

2 4 4 4 8 16

8 16 4 32 64 4 32 64

4 12 32 64

4 12 32 64

f x x x x x x x

x x x x x x x x

x x x x

f x x x x x



Find the asymptotes and intercepts of the function and graph it.

69. 2

2

1

1

x xf x

x

Vertical asymptotes : 1, 1

Horizontal asymptote : 1

intercept : none

intercept : 1

x x

y

x

y y

x

y



Find the intercepts and analyze and graph the rational function.

73. 3 2 2 5

2

x x xf x

x

Vertical asymptote : 2

No horizontal asymptotes

intercept : 2.552

5intercept :

2

x

x x

y y

x

y



Solve the equation algebraically and support graphically.

77. 3 22 3 17 30 0x x x

2

3 2 3 17 30

6 27 30 2 9 10 0

2 9 10 0

2 5 2 0 3 2 5 2 0

2.5 2.5 2 2 3 3

, 2.5 2,3

x x

x x x x x

x x x x




77. 3 22 3 17 30 0x x x

, 2.5 2,3




81. 22 1 3 0x x

1 1 13 3 3

2 2 20 0

1,2

x x x x x



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.



85. a. Find an equation that models the volume of the box.

b. Determine x so that the box has volume 5800 in3.

70 2 30 2V lwh x x x

4.567 in or 8.627 inx x



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


Homework, 9Chapter Review

89. a. Find a linear regression model, and graph it together with a scatter plot of the data.

1.401 4.331y x



89. b. Find a quadratic regression model and graph it with a scatter plot of the data.

20.1875 1.411 13.331y x x



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).



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.



93. a. Express the concentration C (x) of the new mixture as a function of x.

50

50C x

x



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.

33.3 oz.x



93. c. Solve (b) algebraically.50

0.6 50 30 0.6 0.6 20 33.3 oz.50

x x xx


Chapter Test

1. Write an equation for the linear function satisfying the given condition:

( 3) 2 and (4) 9.

2. Write an equation for the quadratic function whose graph contains the

vertex ( 2, 3) and the point

f

f f

(1,2).

3. Write the statement as a power function equation. Let be the constant

of variation. The surface area of a sphere varies directly as the square of

the radius .

4. Divide ( ) by ( ), an

k

S

r

f x d x3 2

d write a summary statement in polynomial form:

( ) 2 7 4 5; ( ) 3

5. Use the Rational Zeros Theorem to write a list of all potential rational

zeros. Then determine which ones, if any, are zero

f x x x x d x x

4 3 2

s.

( ) 2 4 6f x x x x x


Chapter Test

4 3 2

3 2

2

6. Find all zeros of the function. ( ) 10 23

7. Find all zeros and write a linear factorization of the function.

( ) 5 24 12

8. Find the asymptotes and intercepts of the function.

( )

f x x x x

f x x x x

xf x

2

1

19. Solve the equation or inequality algebraically.

122 11

x

x

xx


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

seconds aft

t

er 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?


Chapter Test Solutions

1. Write an equation for the linear function satisfying the given condition:

( 3) 2 and (4) 9.

2. Write an equation for the quadratic function whose graph contains the

vertex ( 2, 3

5

) a

nd

f

f f y x

2 the point (1,2).

3. Write the statement as a power function equation. Let be the constant

of variation. The surface area of a sphere varies directly as the square of

the radi

5 / 9( 2

us

3

)

.

k

r

x

S

y

3 2

2

2

4. Divide ( ) by ( ), and write a summary statement in polynomial form:

( ) 2 7 4 5; ( ) 3

5. Use the Rational Zeros Theorem to write a list of all potential ration

2 2 1

3

f x d x

f x x x x

s kr

x xx

d x x

4 3 2

al

zeros. Then determine which ones, if any, are zeros.

( 1, 2, 3, 6 1/ 2, 3 / 2; 3 / 2 an) 2 4 d 26 f x x x x x



4 3 2

3 2

6. Find all zeros of the function. ( ) 10 23

7. Find all zeros and write a linear factorization of the function.

( ) 5 24 12

8. Find the asymptotes and in

0,5 2

4

tercepts

/ 5, 2 7

of

f x x x x

f x x x x

2

2-intercept (0,1), -intercept none

Vertical Asymptote 1, 1, H

the function.

1( )

1

9. Solve the equation or inequality

orizontal Asymptote

algebraically.

122

1

3/ 2 or 411

x xy x

x x

fx

xx x

x

x

y



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

seconds aft

t2er it is launched.

(b) What is the maximum height of the rock?

(c)

-16 170 6

457.5625ft

5.When will it reach that height?

(d) When will the rock hit the groun

3125

d?

sec

10.66 sec

h t t

homework, page 269 chapter review

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