Chapter 1

[Pages:40]Chapter 1

Chapter 1 Maintaining Mathematical Proficiency (p. 1)

1. 5 23 + 7 = 5 8 + 7 = 40 + 7 = 47

2. 4 - 2(3 + 2)2 = 4 - 2(5)2 = 4 - 2(25) = 4 - 50 = -46

3. 48 ? 42 + --35 = 48 ? 16 + --35 = 3 + --35 = --155 + --35 = --158 = 3--35

4. 50 ? 52 2 = 50 ? 25 2 = 2 2

= 4

5. --12 (22 + 22) = --12 (4 + 22) = --12 (26) = 13

6. --16 (6 + 18) - 22 = --16 (24) - 22 = --16 (24) - 4 = 4 - 4

= 0

7.

y

(0, 6) (3, 6)

4 2

-4 -2

(0, 0) (3, 0) x

8.

y

(0, 4)

(1, 3) (-2, 2) 2

-4 -2

2 4x

9.

y

(-4, 5) (-2, 5)

(-5, 4)

(-1, 4) 2

-6 -4 -2

x

10. Sample answer: Consider the expression 50 ? 25 2. When the order of operations are followed (multiply and divide from left to right), the expression gives 50 ? 25 2 = 2 2 = 4. However, when they are not followed, then the result might be 50 ? 25 2 = 50 ? 50 = 1. Following the order in which transformations are given is also important. For example, translating the point (3, 2) up 3 units and then reflecting in the x-axis, the new coordinate is (3, -5). Reflecting in the x-axis and then translating up 3 units, the new coordinate is (3, 1).

Chapter 1 Mathematical Practices (p. 2)

1.

10

4

-10

10

-6

6

-10

-4

The square viewing window makes the line look steeper.

2.

10

10

-10

10

-11

7

-10

-2

The square viewing window makes the graph look narrower.

3.

10

2

-6

6

-10

10

-10

-6

The square viewing window makes the parabola look wider.

4.

10

5

-10

10

-1

8

-10

-1

The standard viewing window makes the curve look flatter.

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Algebra 2

1

Worked-Out Solutions

Chapter 1

5.

10

2

-6

6

-10

10

-10

-6

The square viewing window makes the curve look wider.

6.

10

4

-10

10

-6

6

-10

-4

The square viewing window makes the curve look wider.

7. -- HWeiidgthht = -- 88 -- ((--82)) = -- 1106 = --58 no; The height-to-width ratio is 5 to 8.

8. -- HWeiidgthht = -- 88 -- ((--72)) = -- 1105 = --23 yes; The height-to-width ratio is 2 to 3.

9. -- HWeiidgthht = -- 98 -- ((--62)) = -- 1105 = --23 yes; The height-to-width ratio is 2 to 3.

10. -- HWeiidgthht = -- 23 -- ((--23)) = --64 = --32 no; The height-to-width ratio is 3 to 2.

11. -- HWeiidgthht = -- 53 -- ((--34)) = --69 = --23 yes; The height-to-width ratio is 2 to 3.

12. -- HWeiidgthht = -- 43 -- ((--43)) = --68 = --34 no; The height-to-width ratio is 3 to 4.

1.1 Explorations (p. 3) 1. a. absolute value; The graph is a "V" shape. b. square root; The domain is x 0 and the range is y 0. c. constant; The y-value is the same for all x-values. d. exponential; The function grows faster as x increases. e. cubic; The domain and range are both all real numbers, and the function grows quickly indicating an exponent. f. linear; The function is always increasing at the same rate. g. reciprocal; The function has a vertical asymptote at the y-axis, and the y-values become smaller as x increases. h. quadratic; The graph is a parabola.

2. Sample answer: Most of the parent functions go through the origin. Most are either symmetric about the y-axis or the origin.

3. a. The equation for the graph is y = x .

b. The equation for the graph is y = --x. c. The equation for the graph is y = 1. d. The equation for the graph is y = 2x. e. The equation for the graph is y = x3. f. The equation for the graph is y = x.

g. The equation for the graph is y = --1x. h. The equation for the graph is y = x2.

1.1 Monitoring Progress (pp. 4 ?7)

1. The function g belongs to the family of quadratics. The graph is translated right and is wider than the graph of the parent quadratic function. The domain of each function is all real numbers and the range of each function is y 0.

2.

y

6

4 g(x) = x + 3

2

y = x

-6 -4 -2 -2

2 4 6x

-4

-6

So, the graph of g(x) = x + 3 is a translation 3 units up of the parent linear function.

3.

y

8

4 y = x2

2

-4 -2 -2

h(x) = (x - 2)2 2 4x

So, the graph of h(x) = (x - 2)2 is a translation 2 units right of the parent quadratic function.

2

Algebra 2

Worked-Out Solutions

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Chapter 1

4.

y

4

2

-4 -2 -2

y= x 2 4x

n(x) = - x

-4

So, the graph of n(x) = x is a reflection in the x-axis of the

parent absolute value function.

5.

y

4

2

y = x

-4 -2

2 4x

g(x) = 3x

So, the graph of g(x) = 3x is a vertical stretch of the parent linear function.

6.

y

10

8

h(x)

=

3 2

x

2

6

4 y = x2

2

-4 -2

2 4x

So, the graph of h(x) = --32 x2 is a vertical stretch of the parent quadratic function.

7.

y

8

y= x

6

4 c(x) = 0.2 x

2

-6 -4 -2

2 4 6x

So, the graph of h(x) = 0.2x is a vertical shrink of the

parent absolute value function.

8.

11

y = x

-3

11

-3

h(x)

=

-

1 4

x

+

5

The graph of h(x) = ---14 x + 5 is a reflection in the x-axis

followed by a vertical shrink and a translation 5 units up of

the parent linear function.

9.

11

y = x2

-5

9

-3 d(x) = 3(x - 5)2 - 1

The graph of h(x) = 3(x - 5)2 - 1 is a translation 5 units right followed by a vertical stretch and then a translation 1 unit down of the parent quadratic function.

10. y

16

12

8

4

0 0 10 20 30 40 50 x

The data appear to lie on a line. So, you can model the data with a linear function. The graph shows that the tank will be empty after 50 minutes.

1.1 Exercises (pp. 8 ?10) Vocabulary and Core Concept Check

1. The function f (x) = x2 is the parent function of f (x) = 2x2 - 3.

2. The question that is different is "What are the vertices of the figure after a translation of 6 units up, followed by a reflection in the x-axis?" The coordinates are: (1, -2) (1, -4) (3, -2) (3, -4) (3, -4) (3, -2) (1, -4) (1, -2) The other three questions result in the coordinates: (1, -2) (3, 4) (3, -2) (5, 4) (3, -4) (5, 2) (1, -4) (3, 2)

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Algebra 2

3

Worked-Out Solutions

Chapter 1

Monitoring Progress and Modeling with Mathematics

3. The function f belongs to the family of absolute value

functions. The graph of f (x) = 2x + 2 - 8 is a horizontal

translation 2 units left followed by a vertical stretch and a vertical translation 8 units down of the parent absolute value function. The domain of each function is all real numbers, but the range of f is y -8, and the range of the parent function is y 0.

4. The function f belongs to the family of quadratic functions. The graph of f (x) = -2x2 + 3 is a reflection in the x-axis followed by a vertical stretch and a vertical translation 3 units up of the parent absolute value function. The domain of each function is all real numbers, but the range of f is y 3, and the range of the parent quadratic function is y 0.

5. The function f belongs to the family of linear functions. The graph of f (x) = 5x - 2 is a vertical stretch followed by a vertical translation 2 units down of the parent linear function. The domain and range of each function is all real numbers.

6. The function f belongs to the family of constant functions. The graph of f (x) = 3 is a vertical translation 2 units up of the parent constant function. The domain of each function is all real numbers, but the range of f is y = 3, and the range of the parent function is y = 1.

Temperature (?F)

7.

y

60

56

52

48

44

y = 2x + 43

40

0 0 2 4 6x

Time (hours)

The type of function that can model the data is a linear function. The temperature is increasing by the same amount at each interval.

8. The type of function that is used as a model is a quadratic function.

9.

y

g(x) = x + 4 4

2

y = x

-2 -2 -4

2 4x

So, the graph of g(x) = x + 4 is a vertical translation 4 units up of the parent linear function.

10.

y

4

y = x

-4

4x

-4

f(x) = x - 6

So, the graph of f (x) = x - 6 is a vertical translation 6 units down of the parent linear function.

11.

y

8

6

4 y = x2 2

-2

2x

-2 f(x) = x2 - 1

So, the graph of f (x) = x2 - 1 is a vertical translation 1 unit down of the parent quadratic function.

12.

y

8

y = x2

6

4

2

-6 -4 -2 h(x) = (x + 4)2 -2

2x

So, the graph of h(x) = (x + 4)2 is a horizontal translation 4 units left of the parent quadratic function.

13.

y

y = x

4

2 -4 -2

g(x) = x - 5 2 4 6 8 10 x

So, the graph of g(x) = x - 5 is a horizontal translation

5 units right of the parent absolute value function.

4

Algebra 2

Worked-Out Solutions

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Chapter 1

14.

y f(x) = 4 + x

10

8

6

4

2

y = x

-6 -4 -2

2 4 6x

So, the graph of f (x) = 4 + x is a vertical translation

4 units up of the parent absolute value function.

15.

y

8

y = x2 4

-4

4x

h(x) = -x2

-8

So, the graph of h(x) = -x2 is a reflection in the x-axis of the parent quadratic function.

16.

y

4

2

y = x

-4 -2 -2

2 4x

-4 g(x) = -x

So, the graph of g(x) = -x is a reflection in the x-axis of the parent linear function.

17.

y 4

f(x) = 3

2

y = 1

-4 -2 -2

2 4x

So, the graph of f (x) = 3 is a vertical translation 2 units up of the parent constant function.

18.

y 2

y = 1

-4 -2

2 4x

-3

f(x) = -2

So, the graph of f (x) = -2 is a vertical translation 3 units down of the parent constant function.

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19.

y

y = x

4

2

f(x)

=

1 3

x

-4

-2

2 4x

-4

So, the graph of f (x) = --13 x is a vertical shrink of the parent linear function.

20.

y

4

2

y = x

-4 -2

2 4x

g(x) = 4x

-5

So, the graph of g(x) = 4x is a vertical stretch of the parent linear function.

21.

y

8

6

4 y = x2 2

-2

f(x) = 2x2 2x

So, the graph of f (x) = 2x2 is a vertical stretch of the parent quadratic function.

22.

y

8

y = x2

6

4

2 -6 -4 -2

h(x)

=

1 3

x2

2 4 6x

So, the graph of h(x) = --13 x2 is a vertical shrink of the parent quadratic function.

Algebra 2

5

Worked-Out Solutions

Chapter 1

23.

-4

y y = x

4

2

h(x)

=

3 4

x

2 4x

-2

-4

So, the graph of h(x) = --34 x is a vertical shrink of the parent linear function.

24.

y

g(x4)

=

4 3

x

2

y = x

-4 -2 -2 -4

2 4x

So, the graph of g(x) = --43 x is a vertical stretch of the parent linear function.

25.

y

4

h(x) = 3x -4 -2

y = x 2 4x

So, the graph of h(x) = 3x is a vertical stretch of the parent

absolute value function.

26.

y

y = x

4

2 -4 -2

f(x) = 12x 2 4x

So, the graph of f (x) = --12 x is a vertical shrink of the parent

absolute value function.

27.

5

f(x) = 3x + 2

-7

y = x 7

-5

So, the graph of f (x) = 3x + 2 is a vertical stretch followed by a vertical translation 2 units up of the parent linear function.

6

Algebra 2

Worked-Out Solutions

28.

8

h(x) = -x + 5 -7

y = x 11

-4

So, the graph of h(x) = -x + 5 is a reflection in the x-axis followed by a vertical translation 5 units up of the parent linear function.

29.

6

-9 h(x) = -3|x| - 1

y = |x| 9

-6

So, the graph of h(x) = -3x - 1 is a reflection in the

x-axis followed by a vertical stretch and then a vertical translation 1 unit down of the parent absolute value function.

30.

4

-6

f(x)

=

3 4

|x|

+

1

y = |x| 6

-4

So, the graph of f (x) = --34 x + 1 is a vertical shrink followed

by a vertical translation 1 unit up of the parent absolute value function.

31.

10

y = x2

-12

12

g(x)

=

1 2

x2

-

6

-7

So, the graph of g(x) = --12 x2 - 6 is a vertical shrink followed by a vertical translation 6 units down of the parent quadratic function.

32.

8

y = x2

-9

9

f(x) = 4x2 - 3

-4

So, the graph of f (x) = 4x2 - 3 is a vertical stretch followed by a vertical translation 3 units down of the parent quadratic function.

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Chapter 1

33.

7

y = x2

-10

10

f(x)

=

-(x

+

3)2

+

1 4

-7

So, the graph of f (x) = -(x + 3)2 + --14 is a translation 3 units left followed by a reflection in the x-axis and then a vertical translation --14 unit up of the parent quadratic function.

34.

3

y = |x|

-4

4

g(x)

=

-|x

-

1|

-

1 2

-3

So, the graph of g(x) = - x - 1 - --12 is a translation 1 unit

right followed by a reflection in the x-axis and then a vertical translation --12 unit down of the parent absolute value function.

35. The error is there is no vertical shrink of the parent quadratic function. The graph is a reflection in the x-axis followed by a vertical stretch of the parent quadratic function.

36. The error is that a translation to the right of 3 units is

represented by a subtraction of 3 in the expression for the

function. The graph is a translation 3 units right of the parent

absolute function, so the function is f (x) = x - 3.

37. (x, y) (x, y - 2) A(2, 1) A(2, -1) B(-1, -2) B(-1, -4) C(2, -3) C(2, -5)

38. (x, y) (x, -y) A(-1, 3) A(-1, -3) B(1, 3) B(1, -3) C(-1, 1) C(-1, -1) D(-3, 1) D(-3, -1)

39. Function g is in the family of absolute value functions. The domain is all real numbers and the range is y -1.

40. Function h is in the family of absolute value functions. The domain is all real numbers and the range is y 2.

41. Function g is in the family of linear functions. The domain is all real numbers and the range is all real numbers.

42. Function f is in the family of linear functions. The domain is all real numbers and the range is all real numbers.

43. Function f is in the family of quadratic functions. The domain is all real numbers and the range is y -2.

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44. Function f is in the family of quadratic functions. The domain is all real numbers and the range is y 6.

45. The type of function that can model the data is an absolute value function because the data are linear and there are positive speeds for the positive and negative displacements. The speed of the car 20 yards past the intersection is estimated to be 8 miles per hour.

46. Sample answer:

y 4

2 y = x2

-4 -2 -4

2 4x y = -x2 - 1

The transformation of the parent quadratic function is a reflection in the x-axis followed by a vertical translation 1 unit down.

47.

4 y f(x) = x - 4

2

-2 -2

2 4 6x

-4 g(x) = x - 4

The graphs are not equivalent. The graph of f is a translation to the right, whereas the graph of g is a translation down of the parent absolute value function.

48. a. The graph of g is a vertical shrink because the factor is --12. b. Multiply the y-coordinate of each point of f by -2.

49. Your friend is correct. Shifting the parent linear function down 2 units will create the same graph as shifting it 2 units right.

50. A linear function can be used to model the distance the swimmer travels. If the person has a 10-meter head start, the type of transformation is a vertical translation. The graph will be shifted up 10 units to represent the head start.

51. a. The type of function modeled by the equation is a quadratic function.

b. The value for t when the ball is released from the hand is zero, because 0 seconds have passed.

c. The ball is 5.2 feet above the ground when it is released from the hand, because this corresponds to t = 0.

52. An absolute value function can be used to model the data.

In this situation, the x-intercept represents the number of hours that have passed at the moment the computer has

0 battery life remaining.

Algebra 2

7

Worked-Out Solutions

Chapter 1

53. a. The function f (x) = 2 x - 3 is a vertical translation of the

parent function. The graph will be shifted 3 units down.

b. The function f (x) = (x - 8)2 is a horizontal translation of the parent function. The graph will be shifted 8 units right.

c. The function f (x) = x + 2 + 4 is both a horizontal

translation and vertical translation of the parent function. The graph will be shifted 2 units left and 4 units up.

d. The function f (x) = 4x2 is neither a horizontal translation nor vertical translation of the parent function. The graph will have a vertical stretch.

54. a. f (x) = 3x + 1; The graph will intersect the x-axis at x = ---31.

b. f (x) = 2x - 6 - 2; The graph will intersect the x-axis at

x = 2 and x = 4.

c. f (x) = -1 x2 + 1; The graph will intersect the x-axis at x = 1 and x = -1.

d. f (x) = 0; This is the x-axis.

Maintaining Mathematical Proficiency

55. f (x) = x + 2 -3 =? 1 + 2 -3 =? 3

-3 3

So, (1, -3) is not a solution.

56. f (x) = x - 3 -5 =? -2 - 3

-5 =? 2 - 3

-5 -1

So, (-2, -5) is not a solution.

57. f (x) = x - 3 2 =? 5 - 3 2 = 2

So, (5, 2) is a solution.

58. f (x) = x - 4 8 =? 12 - 4 8 = 8

So, (12, 8) is a solution.

59. To find the x-intercept let y = 0, then solve for x.

y = x

0 = x

x = 0

To find the y-intercept let x = 0, then solve for y.

y = x

y = 0

So, the x-intercept is (0, 0) and the y-intercept is (0, 0).

8

Algebra 2

Worked-Out Solutions

60. To find the x-intercept let y = 0, then solve for x. y = x + 2 0 = x + 2 x = -2 To find the y-intercept let x = 0, then solve for y. y = x + 2 y = 0 + 2 y = 2 So, the x-intercept is (-2, 0) and the y-intercept is (0, 2).

61. To find the x-intercept let y = 0, then solve for x. 3x + y = 1 3x + 0 = 1 3x = 1 x = --13 To find the y-intercept let x = 0, then solve for y. 3x + y = 1 3(0) + y = 1 y = 1

( ) So, the x-intercept is --13, 0 and the y-intercept is (0, 1).

62. To find the x-intercept let y = 0, then solve for x. x - 2y = 8

x - 2(0) = 8 x = 8

To find the y-intercept let x = 0, then solve for y. x - 2y = 8 0 - 2y = 8

-2y = 8 y = -4

So, the x-intercept is (8, 0) and the y-intercept is (0, -4).

1.2 Explorations (p. 11)

1. The graph of y = x + k is a vertical translation of the parent function f (x) = x. If k is positive, the graph is shifted

up k units. If k is negative, the graph is shifted down k units.

2. The graph of y = x + h is a horizontal translation of the parent function f (x) = x. If h is positive, the graph is

shifted right h units. If h is negative, the graph is shifted left h units.

3. The graph of y = -x is a reflection in the x-axis of the parent function f (x) = x.

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