unit overview - hillsborough county public schools · variation combined variation joint variation...

$: Unit Overview - Hillsborough County Public Schools · variation Combined variation Joint variation Complex fraction Discontinuity ... Unit Overview In this unit, students study radical$
Algebra 2 Honors: Series, Exponential and Logarithmic Functions Semester 2, Unit 5: Activity 25

Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 5

Vocabulary:

Square root regression

One-to-one function

Rational function

Horizontal asymptote

Vertical asymptote

Inverse variation

Constant of variation

Combined variation

Joint variation

Complex fraction

Discontinuity

Removable point of discontinuity

Unit Overview In this unit, students study radical and rational functions. They graph these functions and explore transformations. Students find the roots of these functions and learn to identify asymptotes. Students also explore inverse variation, and they solve rational inequalities.

Student Focus

Main Ideas for success in lessons 25-1, 25-2, 25-3, & 25-4

Explore the square root and cube root functions

Graph and transform square root and cube root functions

Solve square root and cube root equations

Example: Lesson 25-1:

Lesson 25-2:

Page 1 of 32

http://hillsboroughfl.springboardonline.org/


Page 2 of 32

Lesson 25-3: Transformations of

Function Transformation

reflection across the x-axis and vertical stretch by a factor of 8

horizontal translation 5 units to the right

vertical translation down 4 units

vertical shrink by a factor of

, horizontal

translation 4 units to the left, vertical translation 6 units down

vertical translation of the parent function 3 units up, horizontal translation 1 unit right, and reflection across the y-axis

Lesson 25-4:

Page 3 of 32

Page 4 of 32


Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 5

Vocabulary:


One-to-one function

Rational function


Vertical asymptote

Inverse variation


Combined variation

Joint variation

Complex fraction

Discontinuity



Student Focus

Main Ideas for success in lessons 26-1, 26-2, & 26-3

Investigate the inverse relationship between roots and powers

Graph and write the inverse of square root, cube root, quadratic, and cubic functions

Example: Lesson 26-1: Example 1:

Graph the inverse of

. Then give the domain and range of both the function and its inverse.

Page 5 of 32



Square Root Regression – the process of finding a square root function that best fits a set of data. Lesson 26-2:

Page 6 of 32

Lesson 26-3: top row: Cube Root Function bottom row: Inverse of cube root function

Page 7 of 32


Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 5

Vocabulary:


One-to-one function

Rational function


Vertical asymptote

Inverse variation


Combined variation

Joint variation

Complex fraction

Discontinuity



Student Focus

Main Ideas for success in lessons 27-1, 27-2, & 27-3

Write and graph rational functions for real-world situations

Determine asymptotic behaviors and analyze features of the graph

Example: Lesson 27-1: Example: Marcia makes jewelry to sell at the artists’ fair. She spends $120 to rent a stall at the fair for the day. Each piece of jewelry costs Marcia $15 in materials. Which function gives her average cost per piece of jewelry for the x pieces of jewelry she sells at the artists’ fair?

a.

b.

c.

d.

Lesson 27-2: Example:

The population of grizzly bears in a remote area is modeled by the function

,

where t = 1 represents the year 2001, t = 2 represents the year 2002, and so on. Graph the grizzly bear population function.

Page 8 of 32



Lesson 27-3:

Example:

Given the function

,

a. Identify any asymptotes of f. Vertical asymptote: x = 3, Horizontal asymptote: y = 1

b. Identify the x- and y- intercepts of f. x-intercept: -2, y-intercept: -2/3

c. Sketch the graph of f.

Page 9 of 32


Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 5

Vocabulary:


One-to-one function

Rational function


Vertical asymptote

Inverse variation


Combined variation

Joint variation

Complex fraction

Discontinuity



Student Focus

Main Ideas for success in lessons 28-1 & 28-2

Solve problems involving inverse variation

Solve problems related to combined variation

Graph equations of inverse variation and transformations of the parent reciprocal function


Page 10 of 32



Lesson 28-2:

Parent function:

Function Transformation

Translated 2 units to the left

Translated 2 units to the right

Translated 2 units up

Translated 2 units down

Page 11 of 32

Page 12 of 32


Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 5

Vocabulary:


One-to-one function

Rational function


Vertical asymptote

Inverse variation


Combined variation

Joint variation

Complex fraction

Discontinuity



Student Focus

Main Ideas for success in lessons 29-1, 29-2, 29-3, & 29-4

Simplify rational expressions using addition, subtraction, multiplication, and division

Graph rational functions


Page 13 of 32



Page 14 of 32

Lesson 29-2:

Page 15 of 32

Lesson 29-3:

Page 16 of 32

Page 17 of 32

Lesson 29-4:

Page 18 of 32

Page 19 of 32

Page 20 of 32


Resources:

SpringBoard- Algebra 2 Online Resources:

Algebra 2 Springboard Text Unit 5 Vocabulary:


One-to-one function

Rational function


Vertical asymptote

Inverse variation


Combined variation

Joint variation

Complex fraction

Discontinuity



Student Focus Main Ideas for success in lessons 30-1 & 30-2

→ Solve rational equations and inequalities algebraically and graphically → Write equations and inequalities to model real-world situations

Example: Lesson 30-1: Example A:

Page 21 of 32

http://hillsboroughfl.springboardonline.org/�

http://hillsboroughfl.springboardonline.org/�

Example B:

Lesson 30-2: Example A:

Page 22 of 32

Page 23 of 32

Name class date

1© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 5 Practice

LeSSon 25-1 1. If f (x) 5 x , which function describes g (x), a

function with a graph that is a translation of f (x) two units to the right and three units down?

A. x 2 31 2 B. x 2 32 1

C. x2 32 D. x 2 32 2

2. Which function has a domain of [3,`)?

A. f (x) 5 x 32 B. f (x) 5 x 3 31 2

C. f (x) 5 x 3 82 1 D. f (x) 5 x 31

3. Find the domain and range of each function.

a. f (x) 5 x 62

b. f (x) 5 x6 3 22 2 1

c. f (x) 5 x5 22

4. Construct viable arguments. In square root functions, why does the graph shift left or right based on the constant that is added to or subtracted from the variable under the radical? How does this affect the domain?

5. Model with mathematics. The velocity of a cruise ship is equal to the square root of the rate of fuel consumption minus 3 units.

a. Write the velocity function of a cruise ship as a function of the fuel consumption and graph the function.

x

y

b. How many units of fuel are needed to get the ship moving? How does the graph show this?

LeSSon 25-2 6. What is the solution of the equation x45 2 3 5 27?

A. x 5 220, x 5 20 B. x 5 20

C. x 5 210 D. none of the above

7. What is the solution of the equation x 301 5 x?

A. x 5 6, x 5 25 B. x 5 6


Algebra 2 Unit 5 Practice

Page 24 of 32

2

Name class date

© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 5 Practice

8. Make sense of problems. The speed of a passenger jet is a function of the rate of fuel consumption by the engine. Given that the speed is 6 units greater than the square root of the fuel consumption minus 5 units, if the speed of the jet is 15 units, how much fuel is consumed?

9. Attend to precision. What is the solution of the equation x3 7 52 5 ? Graph both sides of the equation and label the solution.

x

y

10. Solve each equation.

a. x5 12 62 5

b. x6 2 3 92 1 5

c. x6 12 32 5

LeSSon 25-3 11. Which transformation of f (x) 5 x3 compresses the

graph and translates it 3 units to the right?

A. g (x) 5 0.5(x 2 3)3 B. g (x) 5 (2(x 2 3))3

C. g (x) 5 (x 1 3)3 D. g (x) 5 1.3(x 2 3)3

12. How does the graph of the function f (x) 5 x3

compare to the graph of the function g (x) 5 13

x3?

13. Model with mathematics. The output of a paper-producing plant is a function of the weight of the raw materials used by the plant. The output is 2 times the cube root of the raw materials used minus 2 units. What is the function describing the output of this paper-producing plant?

14. Graph the function f (x) 5 x7 83 2 . What are the domain and range of this function?

x

y

15. Construct viable arguments. The quantity under the radical of a square root function must be greater than or equal to zero. Explain why the quantity under the radical of a cube root can be less than, greater than, or equal to zero.

LeSSon 25-4 16. What is the solution of the equation x5 3 33 2 5 ?

A. x 5 3 B. x 5 6


Page 25 of 32

3

Name class date


17. Solve the equation x5 23 2 5 x6 43 2 .

18. Model with mathematics. The output of a paper-producing plant is a function of the weight of the raw materials used by the plant. The output is 3 times the cube root of the raw materials used minus 2 units. If the plant produced 9 units of paper, what was the weight of the raw material used by the plant?

19. Make sense of problems. What are the coordinates for the point of intersection between the graphs for the two functions f (x) 5 x8 123 1 and g (x) 5 x4 93 1 ?

20. Solve each of the following equations.

a. x 33 2 5 3

b. x5 23 2 5 2

c. x12 83 2 5 4

LeSSon 26-1 21. What is the inverse function of f (x) 5 (x 1 2)2?

A. f 21(x) 5 x 22 B. f 21(x) 5 x 21

C. f 21(x) 5 x

12

D. f 21(x) 5 x 22

22. What is the inverse function of f (x) 5 x ?

A. f 21(x) 5 x2, x [ R, y [ R

B. f 21(x) 5 x2, x $ 0, y [ R

C. f 21(x) 5 x2, x $ 0, y $ 0

D. f 21(x) 5 x2, x [ R, y $ 0

23. Construct viable arguments. How can you verify that f (x) 5 x3 shares the solution (3, 3) with its inverse?

24. Reason abstractly. If the inverse of function f has domain x $ 0 and range y # 0, what can you say about the domain and range of f ?

25. Consider the following table of values.

Input 5 9 14 17 50 90output 2.8 5 6.7 7.5 13.7 18.7

a. Would f (x) 5 2 x 32 appear to be a reasonable model?

b. How can you evaluate the appropriateness of the function without graphing or using technology?

LeSSon 26-2 26. Which of the following has an inverse that is not

a function?

A. f (x) 5 3 x 22 B. f (x) 5 4x12

C. f (x) 5 3x2 D. f (x) 5 x 31

27. Identify the domain and range of the inverse

function of f (x) 5 x16

2

1 4.

28. Reason quantitatively. Is it necessary to restrict the domain of the function f (x) 5 (x 1 2)2 2 4 in order to ensure that the inverse will be a function? If so, what restriction is necessary?

29. Reason abstractly. Do the constants a and b in a function of the form f (x) 5 (x 1 a)2 2 b affect any restrictions on the domain that ensure that the inverse relation is a function? Explain your reasoning.

Page 26 of 32

4

Name class date


30. Graph the function f (x) 5 x3 22 and its inverse. Identify the domain and range of the inverse function.

x

y

LeSSon 26-3 31. Critique the reasoning of others. Stacey claims

that the cubic function f (x) 5 x3 2 4x2 is a one-to-one function. She says she verified this by making sure the function passed the vertical line test. Is she correct? Why or why not?

32. Use appropriate tools strategically. Use technology to graph the function

f (x) 5 x35

3 1

x2

1 7. Observe the graph to

determine if the function is one-to-one.

33. Identify the inverse of f (x) 5 x2 43 2 .

A. f 21(x) 5 (2x 2 4)3 B. f 21(x) 5 (4x 2 2)3

C. f 21(x) 5 (2x 1 4)3 D. f 21(x) 5 x( 4)

2

3 1

34. Identify the inverse of f (x) 5 (3x 2 5)3.

A. f 21(x) 5 x13

53( )1

B. f 21(x) 5 3 x 53 1

C. f 21(x) 5 x3 53 2

D. f 21(x) 5

x5 33

31

35. The parent cubic function f (x) 5 x3 is one-to-one. Is the cubic function f (x) 5 (x 2 4)3 1 3 also one-to-one? Explain your reasoning.

LeSSon 27-1 36. Model with mathematics. Abbi has decided to

offer a 2-hour computer gaming party for her students. She can rent the student center for $45 per hour, and she can have the technology team set up a network for up to 10 computers for a total of $200 for the evening. Compose function p(x) such that x is the number of students and p(x) is the price per student for her to break even on the cost. Note any appropriate restrictions on the domain x.

37. Reason quantitatively. Based on the situation described in Item 36, if Abbi wants to charge no more than $50 per student to cover fixed costs, what is the minimum number of students she needs to have attending the gaming party?

38. Suppose Abbi learns that the student center will offer a group rate of $500 for up to the full capacity of 70 students to use the center for the evening. Determine the function s(x) such that x is the number of students attending, and s(x) is the price per student, given a total fixed cost (regardless of number of students) of $500, and an additional fee of $4.50 per student for pizza and soda. Note any appropriate restrictions on the domain.

39. Use the function from Item 38. Suppose Abbi charges $50 per student. Write and solve the function to determine the number of students that need to attend for Abbi to break even.

Page 27 of 32

5

Name class date


40. Sketch the graphs of the functions (with appropriate restrictions) in Items 36 and 38 on the same graph to illustrate the differences. Which function indicates the greater change in price per student between the minimum and maximum attendance? Why?

x

y

LeSSon 27-2Use this information for Items 41–45. The population of deer in a remote area is modeled by the function

P(b) 5 b

b750 15

22

, where P(b) represents the deer

population as a function of the number of bears, b.

41. Construct viable arguments. Describe the reasonable domain in set notation. Explain your reasoning.

42. Use technology to graph the deer population function and sketch the result here.

x

y

43. Given a local bear population of 10, what is the deer population?

44. Suppose it was determined that the deer population could be increased by installing feeding stations for the bears. If each station s increases the number of deer by 10, how can the function be modified to indicate the additional population in terms of b and s?

45. Reason quantitatively. Suppose you remove the 215b from the numerator of the fraction in the function. What effect would this have on the function’s reasonable domain?

LeSSon 27-3Use this information for Items 46–50. function

n(b) 5 b

b120 5

41

1

is used to model the number of

undamaged apples on an apple tree in terms of the number of birds b in the area.

46. Use appropriate tools strategically. Use technology to graph the rational function

n(b) 5 b

b120 5

41

1, and sketch the result here.

x

y

47. How many undamaged apples are on the tree before the birds start eating them? How can this be determined from the formula? How can this be determined from the graph?

48. Describe the range of the function in Item 46, using set notation.

49. Describe the domain of the function in Item 46. Use set notation and interval notation.

Page 28 of 32

6

Name class date


50. Reason abstractly. The function n(b) 5 b

b120 5

41

1

has a horizontal asymptote at y 5 5. Describe a possible reason for this, in terms of the model.

LeSSon 28-1 51. y varies inversely as x. When x 5 2, y 5 4. Write an

inverse variation equation describing the relationship between x and y.

52. Create a combined variation equation describing the following relationship. y varies inversely with the square root of x and directly with the cube root of z.

53. Reason quantitatively. The gravitational force between two objects (Fg) varies directly with the mass of each object (m1, m2) and a constant g. The gravitational force varies inversely with the square of the distance between the two objects (r).

a. Write the combined variation equation.

b. Gravitational force is measured in units of newtons (N). The gravitational force between two objects that are 2 m apart is 4 N. What is the gravitational force between the same two objects when they are 4 m apart?

54. y varies inversely as x. When x 5 3, y 5 12. If y 5 9, what is x equal to?

A. 36 B. 18

C. 6 D. 4

55. Construct viable arguments. Why is the following statement true or false? “y varies inversely with the cube root of x. If x changes by a factor of 27, then y changes by a factor of 3.”

LeSSon 28-2 56. Which of the following transformations of the

parent function f (x) 5 x1

has a graph that is translated 6 units left and 5 units up when compared to the graph of the parent function?

A. f (x) 5 x

16

51

2 B. f (x) 5 x

161

C. f (x) 5 x

16

52

1 D. f (x) 5 x

16

51

1

57. Write the function that is a transformation of the parent function f (x) 5

x1

and has a graph that is translated 3 units right and 4 units down.

58. Identify the x-intercept, y-intercept, and vertical and horizontal asymptotes for the function

f (x) 5 x

12

41

1 .

59. Critique the reasoning of others. Sadra claims that the domains of all transformations of the

parent function f (x) 5 x1

are the same as the domain of f (x). Do you agree? Why or why not?

60. Model with mathematics. The graph for a function that describes the pressure in a pipeline as a function of the pipeline’s diameter is a transformation of the parent function f (x) 5

x1

and is translated 3 units up and 4 units left. If the diameter of the pipeline is 0.9 units, what is the pressure in the pipeline?

Page 29 of 32

7

Name class date


LeSSon 29-1 61. What are the restrictions on x for

f (x) 5 x

x x9 1

15 14 3

2

2

2

2 2 2?

A. x fi 3 B. x fi 213

,35

2

C. x fi 213

, 3 D. x fi 23, 13

62. Simplify xx x

13 42

2 1

1 2

x xx x

8 4( 3)( 7)

2

2

2 1 and

identify any restrictions on x.

A. x xx x

3 16 163 4

2

2

1 1

1 2, x fi 2

23

B. x x

x3 4

3 2

2 2 1

2 2, x fi 23

C. xx

3 43 2

1

1, x fi 2

23

D. xx

3 43 2

1

1, x fi 24, 2

23

, 1

63. Simplify x xx x

4 212 5 2

2

2

1 2

2 1

x xx x

8 4( 3)( 7)

2

2

2 1 and


64. Critique the reasoning of others. Claire

simplified f (x) 5 x xx x

3 7 202 11 12

2

2

1 2

1 1. She determined

that f (x) 5 x x

x(3 5)( 4)

2 32 1

1 and that the domain

restriction was x fi 232

. Is Claire correct?

If not, what did she do wrong?

65. Attend to precision. Simplify x x x x

x x25 30 9 16 30 10

15 14 3

2 2

2

1 1 2 2 2

1 1 and identify

any restrictions on x.

LeSSon 29-2

66. Simplify x xx x

11 12 24 9 5

2

2

2 1 1

2 1 1

xx

24 52

2 1 and


67. Simplify x

x4 1

3 11

2 1 2

x xx x

10 12 23 2 1

2

2

1 2

2 2 1 and identify

any restrictions on x.

68. Which of the following are factors of the least

common denominator of xx

5 33 1

1

2 2 and

x xx x

16 30 1015 14 3

2

2

2 2 2

1 1?

A. (3x 1 1) B. (x 1 5)

C. (5x 1 3) D. (3x 2 5)

69. Persevere in solving problems. Identify the least common denominator, add the rational expressions, and identify any restrictions on x.

xx2

21

2 1

x xx x

7 6 410 4

2

2

2 1

1

70. Reason quantitatively. Consider xx

2 5( 2)

2

2 1 1

xx

2 5( 2)

1

1.

a. Can this expression be simplified to x

1021

? If so, how? If not, why not?

b. What restrictions exist on x, if any?

LeSSon 29-3 71. What is the vertical asymptote of x

x x6

9 182

2

2 1?

A. x 5 23 B. x 5 3

C. x 5 6 D. x 5 26

Page 30 of 32

8

Name class date


72. Which equation describes a horizontal or vertical

asymptote of xx x

42 5 122

2

2 2?

A. x 5 0 B. y 5 232

C. x 5 4 D. x 5 232

73. Identify any horizontal or vertical asymptotes or

holes in the graph of xx x

23 102

1

1 2.

74. Make use of structure. Examine the factored form of the function to determine any vertical asymptotes or holes in the graph of the function

x xx x x

3 ( 3)( 4)( 3)

2

1 2.

75. express regularity in repeated reasoning. Identify the horizontal asymptotes in each expression.

a. x x xx x x

2 3 52 3 4 5

3 2

3 2

1 1 1

1 1 1

b. x xx x x

2 3 52 3 4 5

2

3 2

1 1

1 1 1

c. x x xx x2 3 5

3 4 5

3 2

2

1 1 1

1 1

d. x xx x

2 3 53 4 5

2

2

1 1

1 1

LeSSon 29-4 76. Which expression describes a horizontal or vertical

asymptote of f (x), as shown in the graph?

x

y

25

25

5

5(0, 21)

A. x 5 21 B. y 5 21

C. x 5 2 D. y 5 0

77. Consider the rational function f (x) 5 x x

34

522

11

.

a. Analyze and sketch the graph of f (x).

x

y

b. Identify any asymptotes or holes.

78. Describe the domain of f (x) 5 x x

34

522

11

using set notation.

Page 31 of 32

9

Name class date


79. Use appropriate tools strategically. Use technology to graph the function

f (x) 5 x x

21

322

21

and determine any horizontal

or vertical asymptotes.

80. Critique the reasoning of others. Evan simplified

the rational function f (x) 5 xx

25

2

1 1

x xx x

5 68 15

2

2

1 1

1 1.

He identified a vertical asymptote at x 5 25 and horizontal asymptotes at y 5 23 and y 5 2. Was he correct? Why or why not?

LeSSon 30-1 81. Solve the equation

x7

3 1

x36

1 x4

5 x7

.

82. What is the solution of the equation

x4

22 1

x1

4 5

x6

?

A. x 5 23

B. x 5 223

C. x 5 467

D. x 5 2467

83. What is the solution of the equation

x3

4 1

x52

1 x1

5 x8

?

84. Construct viable arguments. Two cranes unload ships at a dock. Together they unload 50 tons of cargo every 15 minutes. If the first crane unloads 100 tons of cargo every hour, the second crane also unloads 100 tons of cargo every hour. Is this reasonable? Why or why not?

85. Model with mathematics. Two enzymes, enzyme A and enzyme B, catalyze the same reaction. Enzyme B is three times faster than enzyme A. Together they catalyze the reaction of 10 micrograms of reactant in 3 milliseconds. How long would it take each enzyme to catalyze the reaction of 10 micrograms of reactant on its own?

LeSSon 30-2 86. What is the solution of the inequality x

x8

92 2 $ 0?

A. x , 23 or 0 # x , 3

B. x . 3 or 23 , x # 0

C. 0 # x , 3

D. 0 , x , 3

87. Solve the inequality x3 1

x1

21 $ 0 algebraically or

graphically.

88. Model with mathematics. A refinery was constructed at a cost of $100,000,000. Each gallon of gas is produced at this refinery for an additional cost of $0.50. Write the function representing the total cost per gallon.

89. Make sense of problems. In Item 88, how many gallons of gas does the refinery need to produce to keep the average cost of a gallon of gas under $3?

90. Solve the inequality 9x 2 x1

. 0 algebraically or graphically.

Page 32 of 32

unit overview - hillsborough county public schools · variation combined variation joint variation...

Documents