CHAPTER Solutions Key 5 Quadratic Functions

[Pages:58]CHAPTER Solutions Key 5 Quadratic Functions

ARE YOU READY? PAGE 311

1. E 3. A 5. (3.2)2

= (3.2)(3.2) = 10.24

7. 121 = 11 9. 72

= 36 ? 2 = 62

11. 33 ? 75 = 2475 = 225 ? 11 = 1511

13. (x 2)(x 6) = x2 6x 2x + 12 = x2 8x + 12

15. (x + 2)(x + 7) = x2 + 7x + 2x + 14 = x2 + 9x + 14

2. C

4. B

( ) 6. _2_ 2 5 = ( _2_ )( _2_ ) = _4_5_ 5 25

8. _1__ = _1_

16 4

10. 2(144 4) = 2(12 4) = 2(8) = 16

12. __5_4_

3 = 18 = 9 ? 2 = 32

14. (x + 9)(x 9) = x2 9x + 9x 81 = x2 81

16. (2x 3)(5x + 1) = 10x2 + 2x 15x 3 = 10x2 13x 3

5-1 USING TRANSFORMATIONS TO GRAPH QUADRATIC FUNCTIONS, PAGES 315-322

CHECK IT OUT!

1. x

g(x) = x2 + 6x 8

1 g(1) = (1)2 + 6(1) 8 = 3

2 g(2) = (2)2 + 6(2) 8 = 0

3 g(3) = (3)2 + 6(3) 8 = 1

4 g(4) = (4)2 + 6(4) 8 = 0

5 g(5) = (5)2 + 6(5) 8 = 3

y

x

(x, g(x))

(1, 3) (2, 0) (3, 1) (4, 0) (5, 3)

2a. g is f translated 5 units down.

y

x

17. 2x + 10 = 32 2x = 32 10 2x = 42 x = 21

19. _2_(x 1) = 11

3_23_x

_2_ = 11 _2_3x = _3_5_

3 x

=

_13_0_5_

x = 176_1_

2

21.

y

x

18. 2x (1 x ) = 2 2x 1 + x = 2 3x = 2 + 1 3x = 3 x=1

20. 2(x + 5) 2x + 10

5x = 1 5x = 1 3x = 9

x=3

22.

y

x

23.

y

24.

y

x

x

b. g is f ranslated 3 units left and 2 units down.

y

x

3a. g is a horizontal compression of f by a factor of _1_.

2

y

x

b. g is f reflected across the x-axis and vertically

compressed by a factor of _1_.

2

y

x

153

Holt McDougal Algebra 2

4a. g(x) = _1_(x 2)2 4

3

b. g(x) = (x + 5)2 + 1

5. _d_n_(_v_) = _0_.0_3_9__v_2 = _1_3_

d(v) Vertical

0.045v2 15 compression by

a

factor

of

_1_3_;

the

braking

15

distance wiil be less with optimally inflated new tires

than with tires having more wear.

THINK AND DISCUSS

1. Possible answer: a indicates a reflection, vertical stretch or vertical compression. h indicates a horizontal translation (left or right). k indicates a vertical translation (up or down).

2. Possible answer: The function for which a is greater will have a narrower graph.

3. 4RANSFORMATION %QUATION

6ERTICAL TRANSLATION

f x x

'RAPH

y

x

(ORIZONTAL TRANSLATION

f x x

y

x

2EFLECTION

f x x

y x

6ERTICAL STRETCH

f x x

y

6ERTICAL COMPRESSION

f x ?x

x

y

x

EXERCISES

GUIDED PRACTICE

1. vertex

2. x

2

1

0

1

2

f(x)

12

6

4

6

12

y x

3. x

1

0

1

2

3

g(x)

6

2

0

0

2

y

x

4. x

2

1

0

1

2

h(x)

0

y

1

0

3

8

x

5. d is f translated 4 units right.

y

x

6. g is f translated 3 units right and 2 units up.

y

x

7. h is f translated 1 unit left and 3 units down.

y

x

154

Holt McDougal Algebra 2

8. g is a vertical stretch of f by a factor of 3.

y

PRACTICE AND PROBLEM SOLVING

17. x

2

1

0

1

2

x

9. h is a horizontal stretch of f by a factor of 8.

y

x

f(x)

0

3

4

3

0

y

x

18. x

2

1

0

1

2

10. p is a vertical compression of f by a factor of 0.25.

y

x

g(x)

9

4

1

0

1

y

x

11. h is f reflected across the x-axis and horizontally

compressed by a factor of _1_.

5

y

x

12. g is a vertical stretch of f by a factor of 4.2.

y

19. x

2

1

0

1

2

h(x)

1

3

1

5

15

y

x

x

13. d is f reflected across the x-axis and vertically

compressed by a factor of _2_.

3

y

x

14. g(x) = 2(x + 3)2

15. h(x) = x2 6

16. Vertical compression by a factor of _4_1_5_; the safe

592 working load is less for an old rope than for a newer

rope of the same radius.

20. g is f translated 2 units down.

y

x

21. h is f translated 5 units left.

y

x

22. j is f translated 1 unit right.

y

x

155

Holt McDougal Algebra 2

23. g is f translated 4 units left and 3 units down.

y

x

24. h is f translated 2 units up and 2 units left.

y

x

25. h is f translated 4 units right and 9 units down.

y

x

26. g is a vertical compression of f by a factor of _74_.

y

x

27. h is f reflected across the x-axis and vertically

stretched by a factor of 20.

y

x

28. j is a horizontal stretch of f by a factor of 3.

y

x

29. g(x) = _21_(x 1)2

30. h(x) = 2.5(x + 2)2 + 1

31. Vertical translaton; at any given speed, the gas mileage for an SUV is 18 mi/gal less than for a compact car.

32a. Translation 10 units right and 300 units up.

b. No; the largest pen Keille can build with an 80 ft roll has an area of 400 ft2, and the largest pen she can build with a 40 ft roll has an area of 100 ft2. Therefore, a roll that is twice as long allows her to build a pen with 4 times the area.

33. p is f reflected across the x-axis and translated 4 units right.

34. g is f vertically stretched by a factor of 8 and translated 2 units left.

35. h is f vertically stretched by a factor of 4 and

translated 2 units down.

36.

p is f vertically compressed translated 2 units up.

by

a

factor

of

_1_

4

and

37.

g is f horizontally compressed translated 1 unit up.

by

a

factor

of

_1_

3

and

38. h is f reflected across the x-axis and horizontally stretched by a factor of 3.

39. C

41. A 42a. B(r) = 25 r2

40. B

b. B is A reflected across the x-axis and translated 25 units up.

c. A: D: {r | 0 r 2.5}; R: {A | 0 A 6.25}.

B: D: {r | 0 r 2.5}; R: {B | (25 6.25) B 25}.

Possible answer: The radius of the circle cannot be

less than 0 or greater than half the side length of

the square.

43. Horizontal line; linear or constant function.

44. Very narrow parabola opening upward with its vertex at ( 5, 5).

45a. Vertical compression by a factor of 0.38 and translation 2.5 units right and 59 units up.

b. y = 6.08(t 4)2 + 95

TEST PREP

46. B 48. C 50. 5

47. J 49. G

CHALLENGE AND EXTEND

51. y = 3(x 3)2 + 3

Translation 6 units up and 6 units right.

52a.

f: horizontal compression translation 2 units down.

by

a

factor

of

_1_

2

and

g: vertical stretch by a factor of 4 and translation 2

units down.

b.

y

x

c. The functions are the same. d. g(x) = (3x)2

SPIRAL REVIEW

53. Yes; the price is justified because the volume of the

large container is more than 3 times the volume of

the small container: Vsmall = 28; Vlarge = 88.

54. f(x) = x

55. f(x) = x

156

Holt McDougal Algebra 2

56. 2y + 5x = 14

2y = 5x + 14

y = _25_x + 7

y

x

57. x

_21_y

+

4 =

_21_y =

1 x

5

y = 2x + 10

y

x

5-2 PROPERTIES OF QUADRATIC FUNCTIONS IN STANDARD FORM, PAGES 323-330

CHECK IT OUT!

1. x = 3

2a. (a) downward

(b)

x

=

__b_

2a

=

__(___4_)

2( 2)

=

_4__

4

=

1

(c) f( 1) = 2( 1)2 4( 1) = 2(1) + 4 = 2+4 = 2

The vertex is ( 1, 2).

(d) The y-intercept is 0.

(e)

y

x

b. (a) upward

(b)

x

=

__b_

2a

=

___3_

2(1)

=

_3_

2

( ) ( ) ( ) (c) g

_3_ =

2

_3_

2

2

+ 3

_3_

2

1

= _9_ _9_ 1

4 2

= _1_3_

( ) 4

The vertex is

_23_,

_1_3_

4

.

(d) The y-intercept is 1.

(e)

y

x

3a.

x

=

__b_

2a

=

__(___6_)

2(1)

=

_6_

2

=

3

b.

x

=

__b_

2a

=

__0___

2( 2)

=

0

f(3) = (3)2 6(3) + 3

g(0) = 2(0)2 4

= 9 18 + 3 =6 The minimum is 6.

D: ; R: {y | y 6}.

=4 The maximum is 4.

D: ; R: {y | y 4}.

4.

s

=

__b_

a

=

____2_._4_5__

2( 0.025)

=

__2_._4_5_

0.05

=

49

m(49) = 0.025(49)2 + 2.45(49) 30

= 0.025(2401) + 120.05 30

= 30.025

The maximum mileage is 30.0 mi/gal at a speed of

49 mi/h.

THINK AND DISCUSS

1. Possible answer: No; quadratic functions open in 1 direction. If they open upward, they have a minimum value. If they open downward, they have a maximum value.

2. Possible answer: The value of x2 increases faster than the value of 2x decreases.

3.

/PENSUPWARDOR DOWNWARDUPWARDIFa ISPOSITIVEANDDOWNWARD

? !XISOFSYMMETRY xab

IFaISNEGATIVE

0ROPERTIES

y INTERCEPTc

? ?

OF0ARABOLAS

6ERTEX ba f ba

EXERCISES

GUIDED PRACTICE

1. minimum

2. x = 2

3. x = 0 5a.downward

4. b.

x x

= =

__5b_

=

__(___2_)

=

1

2a 2( 1)

c. f( 1) = ( 1)2 2( 1) 8

= 1+2 8

=7

The vertex is ( 1, 7).

d. The y-intercept is 8.

e.

y

x

157

Holt McDougal Algebra 2

6a. upward

7a. downward

b.

x

=

__b_

2a

=

__(___3_)

2(1)

=

_3_

2

b.

x

=

__b_

2a

=

___4__

2( 1)

=

2

( ) ( ) ( ) c.

g

_3_

2

=

_3_

2

2

3 _3_ + 2 c. h(2) = (2)2 + 4(2)

2

= 4+8 1

1

= _9_ _9_ + 2

= 3

= 4 _1_ 2

4

The vertex is (2, 3). d. The y-intercept is 1.

( ) The vertex is _32_,

_1_

4

.

e.

y x

d. The y-intercept is 2.

e.

y

x

8.

x

=

__b_

2a

=

__0__

2(1)

=

0

9.

x

=

__b_

2a

=

___3__

2( 1)

=

_3_

2

( ) ( ) ( ) f(0) = 1

The minimum is 1.

g _3_ = _3_ 2 + 3 _3_ 2

2

2

2

D: ; R: {y | y 1}.

= _9_ + _9_

42

2

= _1_

The

4 maximum

is

_14_.

10.

x

=

__b_

2a

=

___3_2__

2( 16)

=

1

D: R:

; {y |

y

_41_}.

h(1) = 16(1)2 + 32(1) + 4

= 16 + 32 + 4

= 20

The maximum is 20.

D: ;

R: {y | y 20}.

11.

x

=

__b_

2a

=

____0_._2_5__

2( 0.005)

=

25

h(25) = 0.005(25)2 + 0.25(25)

= 3.125 + 6.25

= 3.125

The maximum height is 3.125 m.

PRACTICE AND PROBLEM SOLVING

12. x = 0 14. x = 1

13. x = 1

15a. upward

b.

x

=

__b_

2a

=

___1_

2(1)

=

_1_

2

( ) ( ) ( ) c. f

_1_ =

2

_1_ 2 +

2

_1_

2

2

= _1_ _1_ = 4 _9_ 2

2

4

( ) The vertex is

_12_,

_9_

4

.

d. The y-intercept is 2.

e.

y

x

16a. downward

b.

x

=

__b_

2a

=

___6__

2( 3)

=

1

c. g(1) = 3(1)2 + 6(1) = 3+6 = 3

The vertex is (1, 3).

d. The y-intercept is 0.

e.

y

x

17a. upward

b.

x

=

__b_

2a

=

__(___2_)

2(0.5)

=

2

c. h(2) = 0.5(2)2 2(2) 4

= 0.5(4) 4 4

=6

The vertex is (2, 6).

d. The y-intercept is 4.

e.

y

x

158

Holt McDougal Algebra 2

18a. downward

b.

x

=

__b_

2a

=

___8__

2( 2)

=

2

c. f(2) = 2(2)2 + 8(2) + 5 = 8 + 16 + 5 = 13

The vertex is (2, 13).

d. The y-intercept is 5.

e.

y

x

19a. upward

b.

x

=

__b_

2a

=

___2_

2(3)

=

_1_

3

( ) ( ) ( ) c. g

_1_

3

= 3

_1_

3

2

+

2

_1_

3

8

= _1_ _2_ = 3 _2_5_ 3

8

( ) 3

The vertex is

_13_,

_2_5_

3

.

d. The y-intercept is 8.

e.

y

x

20a. upward

b.

x

=

__b_

2a

=

___2_

2(1)

=

1

c. h( 1) = 2( 1) 1 + ( 1)2

= 2 1+1

=2

The vertex is ( 1, 2).

d. The y-intercept is 1.

e.

y

x

21a. downward

b.

x

=

__b_

2a

=

__0___

2( 1)

=

0

c. f(0) = 2 (0)2 = 2

The vertex is (0, 2).

d. The y-intercept is 2.

e.

y

x

22a. upward

b.

x

=

__b_

2a

=

___3__

2(0.5)

=

3

c. g( 3) = 0.5( 3)2 + 3( 3) 5

= 4.5 9 5

= 9.5

The vertex is ( 3, 9.5).

d. The y-intercept is 5.

e.

y

x

23a. upward

( ) b.

x

=

__b_

2a

=

___1_

2

_1_

4

=

2

c. h(

2) = _1_(

4

2)2

2 + 2

=1 2+2

= 1

The vertex is ( 2, 1).

d. The y-intercept is 2.

e.

y

x

24. x = __b_ = ___7__ = _7_ 25. x = __b_ = ___6__ = 3

2a 2( 2) 4

2a 2( 1)

( ) ( ) ( ) f _7_ =

2

_7_

2

+ 7

_7_

4

4

4

3

g(3) = 6(3) (3)2 = 18 9

( ) =

2

_4_9_

16

+ _4_9_

4

3

= 9 The maximum is 9.

D: ;

= _4_9_ + _9_8_ _2_4_

R: {y | y 9}.

888

= 3.125

The maximum is 3.125.

D: ;

R: {y | y 3.125}.

159

Holt McDougal Algebra 2

( ) 26.

x

=

__b_

2a

=

__(___4_)

2(1)

=

2

h(2) = (2)2 4(2) + 3

27.

x

=

__b_

2a

=

___0___

2

_1_

2

=

0

=4 8+3 =1

f(0) =

_1_(0)2

2

4= 4

The minimum is 1.

The maximum is 4.

D: ; R: {y | y 1}.

D: ; R: {y | y 4}.

28.

x

=

__b_

2a

=

__(___6_)

2( 1)

=

3

g( 3) = ( 3)2 6( 3) + 1

= 9 + 18 + 1

= 10

The maximum is 10.

D: ;

R: {y | y 10}.

29.

x

=

__b_

2a

=

___8_

2(1)

=

4

h( 4) = ( 4)2 + 8( 4) + 16 = 16 32 + 16 = 0

The minimum is 0.

D: ;

R: {y | y 0}.

30.

d

=

__b_

2a

=

____0_.6__5_7__

2( 0.0018)

=

182.5

T = 0.0018(182.5)2 + 0.657(182.5) + 50.95

= 59.95 + 119.9 + 50.95

111

The maximum temperature in 2003 is approximately 111?.

31. Maximum height is 64 ft. Possible answer: The axis of symmetry is halfway between any 2 points with the same y-value. Because the points (1, 48) and (3, 48) have the same y-value, the axis of symmetry is x = 2. Because the vertex lies on the axis of symmetry, the vertex of the graph is (2, 64). Therefore, the maximum value of the function is 64.

'OLF"ALL(EIGHT

(EIGHTFT

4IMES

32a. C(x) = x(32 2x)

b. x

0

C(x) 0

#ROSS SECTIONALAREACM

4

8

12

16

96 128 96

0

7IDTHOFBENDCM

c. D: {0 x 16}; R:{y | 0 y 128} Neither the width nor the area can be negative.

d. x = 8 cm

33a. t = __b_ = ___3_0_0_0__ = 0.375

2a 2( 4000) h(0.375) = 4000(0.375)2 + 3000(0.375)

= 562.5 + 1125 = 562.5 mm

b. 93.75 to 1.

Possible answer: The ratio for spittle bugs is more

than 67 times as great as the ratio for humans.

c.

_x__

1.8

=

93.75

x = 168.75 m

34. A(x) = 10x x2

x = __b_ = ___1_0_ = 5

2a 2( 1) A(5) = 10(5) (5)2

= 50 25 = 25 yd2

35. min 3.029771

36. max 13.178533

37. min 1.253333

38. max = 5.3715

39. The axis of symmetry is halfway between any 2 points with the same y-value. Halfway between 7 and 3 is 2. Therefore, the axis of symmetry is x = 2.

40. Yes; possible answer: a function such as f(x) = x2 5 may open downward and have a

vertex below the x-axis. A function such as f(x) = x2 + 2 may open upward

and have a vertex above the x-axis.

41a.

t

=

__b_

2a

=

___5_0__

2( 16)

1.6

s

b. h(1.5625) = 16(1.5625)2 + 50(1.5625) + 6

= 39.0625 + 78.125 + 6

45 ft

TEST PREP

42. C 44. B

43. G 45. G

160

Holt McDougal Algebra 2

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