LESSON 1: LINES AND LINEAR FUNCTIONS EXERCISESrusdmath.weebly.com/uploads/1/1/1/5/11156667/g9_u2_-_exercises_onl… · 11. Lian is taking guitar lessons with a private instructor

High School: Linear Functions

EXERCISES

Copyright © 2015 Pearson Education, Inc. 5

EXERCISES

1. Write what you already know about linear functions.

Share your summary with a classmate. Did you write the same things?

2. Write your wonderings about linear functions.

Share your wonderings with a classmate.

3. Write a goal stating what you plan to accomplish in this unit.

4. Based on your previous work in math, write three things that you will do during this unit to increase your success.

For example, consider ways you will participate in classroom discussions, your study habits, how you will organize your time, what you will do when you have a question, and so on.

LESSON 1: LINES AND LINEAR FUNCTIONS



EXERCISESLESSON 2: INPUT-OUTPUT MACHINES

EXERCISES

1. Which of these sets of inputs/outputs are for the function subtract 2 from the input and then multiply by 3? There may be more than one set of inputs/outputs.

A input: 5, output: 9

B input: 3, output: 3

C input: 10, output: 20

D input: 7, output: 12

E input: 12, output: 30

2. A function adds 5 and then multiplies the sum by 3. Which of these representations describe the function? There may be more than one representation.

A y = 3(x + 5)

B f(x) = 3x + 5

C x → 3x + 5

D x → x + 5 • 3

E f(x) = 3(x + 5)

3. This function is expressed in function notation.

f x x( ) ( )= −1

23

Which of these representations describe this function? There may be more than one representation.

A Subtract 3 from the input and then multiply by 12

.

B Multiply 12

times the quantity written as the input minus 3.

C x x→ 12

32

–

D Multiply 12

times the input and subtract 3.

E y x= −12

3( )

4. A function divides the input by 2 and then adds 4.

Write this function using function notation.




5. Write a description in words for the function f(x) = 2(7 – x).

6. Write the described function using function notation.

For g of x, multiply the input by 4 and subtract 3.

7. A function subtracts 3 from the input and then multiplies by 2.

Describe this function algebraically.

8. The electric company charges $50 per month plus $1.25 per kilowatt.

Write the cost (c) of your monthly bill in terms of hours (h) used.

Write your answer in function, equation, or mapping notation.

9. A function divides the input by 2 and then adds 4.

Complete the table. Write your numbers as decimals.

Input –5 –2 0 2 5

Output

10. A function subtracts 3 from the input and then multiplies by 2.

Make a table containing five inputs and their corresponding outputs.

Input Output

11. Lian is taking guitar lessons with a private instructor. On weekdays, her teacher charges a base fee of $20 plus $0.50 for each minute of instruction.

a. How do you know this relationship represents a function? b. Describe the domain and range of this function.




Challenge Problem

12. Explain each statement.

a. If g(4) = –12, then the point (4, –12) lies on the graph of g.

b. If f(x) = ax + b, then the point (0, b) lies on the graph of f.



EXERCISES

EXERCISES

1. Consider this graph.

–6 –4 –2 2 4 6 x

y

–6

–4

–2

2

4

6

a. Open dots represent of the value.A inclusion B exclusion

b. Closed dots represent of the value.A inclusion B exclusion

c. How many outputs are there for each input in the graph?A zero B one C two D all real numbers

d. Does this graph represent a function?A yes B no

LESSON 3: FUNCTIONS AND NON-FUNCTIONS


EXERCISES


LESSON 3: FUNCTIONS AND NON-FUNCTIONS

2. What are the domain and range of this function?

–6 –4 –2 2 4 6 x

y

–6

–4

–2

2

4

6

3. Does this graph represent a function? Explain how you know.

–6 –4 –2 2 4 6 x

y

–6

–4

–2

2

4

6



EXERCISESLESSON 3: FUNCTIONS AND NON-FUNCTIONS

4. What are some possible outputs if the input is 0?

–6 –4 –2 2 4 6 x

y

–6

–4

–2

2

4

6




5. Which of these graphs represents a function?

A

–6 –4 –2–10 –8 2 4 6 8 10 x

y

–6

–4

–2

–10

–8

2

4

6

8

10 B

–6 –4 –2–10 –8 2 4 6 8 10 x

y

–6

–4

–2

–10

–8

2

4

6

8

10

C

–5 –2.5 2.5 5 x

y

–2.5

–5

2.5

5 D

–6 –4 –2 2 4 6 x

y

–6

–4

–2

2

4

6

6. A function is defined such that the output gives the greatest even integer that is less than or equal to the input.

Complete this table.

Input Output

–3

–2

0

5 4

10




7. A function is defined such that the output gives the greatest even number that is less than or equal to the input.

a. Sketch a graph of this relationship.

b. Does it represent a function? Explain how you know.

8. A function is defined such that the output gives the greatest even number that is less than or equal to the input.

What are the domain and range?

9. Which of these statements describes a property of functions? Select all that apply.

A Functions can be described using words.

B Functions must use numbers as the input and output.

C There is only one output for each input.

D There is only one input for each output.

E Functions can often be written as an expression with a variable for the input.

Challenge Problem

10. The formula for a linear function f is given by f(x) = ax + b.

a. Explain why x = 5 is not a function but can be represented by a straight line.

b. For which values of a and/or b does f approach a vertical line?


EXERCISES


LESSON 4: DETERMINING A LINE

EXERCISES

1. Which of these equations is the equation 2x + 3y = 12 in slope-intercept form?

A y x= +32

6

B y x= +–23

4

C y x + =632

D y x + + =23

4 0

2. –3x + 4y = –9

a. Write this equation in slope-intercept form.

b. Determine the rate of change (slope) and the x- and y-intercepts.

3. What is the equation of the line that goes through points (1, 2) and (3, 6)?

A y = 2x

B y x= +12

32

C y = 2x +1

D y x= 12

4. What is the equation of the line that goes through points (–3, –2) and (3, 6)?

A y x= +23

2

B y x= 23

C y x= +34

14

D y x= +43

2



EXERCISESLESSON 4: DETERMINING A LINE

5. Write the equation of the line that goes through points (–5, 2) and (–1, –5).

6. a. Write the equation of the line that has a y-intercept of 4 and rate of change of –2.

b. Graph the equation.

7. a. Write the equation of the line that has a y-intercept of –2 and slope of 4.


8. a. Write the equation of the line for which f(0) = 2.5 and f(–3) = 4.


9. a. Write the equation of the line that has a slope of 4.3 and contains the point (1, 1).


10. a. Write the equation of the line that has an x-intercept of 3 and a y-intercept of 7.


11. Choose any two points (p, q) and (r, s) such that p, q, r, and s are integers and p ≠ r.

What is the slope of the line through these two points?

12. Choose any two points (p, q) and (r, s) such that p, q, r, and s are integers and p ≠ r.

a. Find the y-intercept.

b. Write the equation of your line in slope-intercept form.

Challenge Problem

13. Consider the equation of the form y = mx + 4.

a. Draw the graph for m = 2.b. Find m if the x-intercept is 4.c. Find m if the graph goes through the point (3, 5).


EXERCISES


EXERCISES

1. Use the points from the table of the original function to fill in the table of values for the inverse function.

Linear function:

x –2 –1 0 1 2

y 6 3 0 –3 –6

Inverse function:

x

y

2. What is the slope of the inverse function of y = 2x – 5?

A − 12

B 12

C 2

D 5

3. Find the inverse function of y = x + 5.

A x = y – 5

B y = x – 5

C y = –x – 5

D y = –x + 5

4. Which is the equation for the inverse function of y x= −45

1?

A y x= −54

5

B y x= +54

5

C y x= −54

54

D y x= +54

54

LESSON 5: INVERSES OF LINEAR FUNCTIONS


EXERCISES


5. Write the equation for the inverse function of y = 0.4x – 1.2.

6. Write the equation for the inverse function of y x= +–72

14 .

7. Look at this graph.

–15 –10 –5 5 10 15 20 x

y

–15

–10

–5

5

10

15

y x= −15

3

Draw the graph of the inverse function.




EXERCISES

8. Does this graph represent a linear function and its inverse? How do you know?

–5 5 10 x

y

–5

5

10

9. Does this graph represent a linear function and its inverse? How do you know?

–5 5 10 x

y

–5

5

10




EXERCISES

10. When does a linear function have 0 points in common with its inverse? infinitely many points? Explain.

Challenge Problem

11. Show that f(f –1[x]) = f –1(f [x]) = x for any linear function f(x) and its inverse f –1(x).



EXERCISES


EXERCISES

1. Which of these functions translates y = 2x + 4 up 7 units?

A y = 2x – 3

B y = 2x + 11

C y = –5x + 11

D y = 9x – 3

2. Which of these equations are translations of y x= − −12

1?

y x= − +12

2 2 5y x+ = 2 8y x− = y x= −12

2

3. Which of these functions intersects the y-axis at the same point as y = 2x + 4?

A y = 2x – 4

B y x= +15

4

C y x= − +12

2

D y = –2x – 4

4. Which of these functions is parallel to y = –x + 4 and passes through point (3, 2)?

A y = –x + 5

B y = x + 5

C y = x + 1

D y = –x – 1

LESSON 6: ALTERING FUNCTIONS



EXERCISESLESSON 6: ALTERING FUNCTIONS

5. a. Draw the graph of f x x( ) = +32

2 .

b. In the same grid, draw the graph of f(x) – 7.c. Explain the relationship between the two graphs in terms of the general form

of a linear function, f(x) = mx + b.

6. a. Draw the graph of f x x( ) = +32

2 .

b. In the same grid, draw the graph of –2f(x). c. Explain the relationship between the two graphs in terms of the general form

of a linear function, f(x) = mx + b.

7. A junior carpenter does small jobs at the homes of friends and relatives at a rate of $40 per hour. Soon, she becomes so popular that she charges $50 up front in addition to the hourly rate.

a. Find the formulas for both scenarios.b. Suppose you have $200 to spend. How long could the carpenter work for you

before and after the up-front fee?c. Draw the graphs for both scenarios in the same coordinate system.

Challenge Problem

8. Let f(x) = mx + b. Show that the x-intercept of g(x) = kf(x) is equal to that of f(x) for any k ≠ 0.



EXERCISES

EXERCISES

1. Consider the function f(x) = 3x + 5. Which of these functions shifts f(x) 4 units down? There may be more than one correct function.

A g(x) = 3x + 9

B g x x( ) –= 13

7

C g(x) = 3x + 1

D f(x) – 4 → g(x)

E f(x – 4) → g(x)

2. Consider the function f(x) = –3x. Which of these functions shifts f(x) 3 units to the right? There may be more than one correct function.

A g(x) = 3x – 9

B g(x) = –3x + 9

C g(x) = –3x + 3

D f(x + 3) → g(x)

E f(x – 3) → g(x)

3. Consider the function f(x) = –4x – 7. Which of these functions shifts f(x) 5 units to the left? There may be more than one correct function.

A g(x) = –4x – 27

B g(x) = –9x – 7

C g x x( ) = −14

7

D f(x + 5) → g(x)

E f(x – 5) → g(x)

4. a. Graph f(x) = 3x − 2 as Line A.

b. Graph a function g(x) that shifts the graph of f(x) 5 units to the left as Line B.

c. Write an equation for your new function g(x), in terms of f(x).

LESSON 7: CHANGING A FUNCTION INPUT



EXERCISESLESSON 7: CHANGING A FUNCTION INPUT

5. a. Graph f(x) = 3x – 2 as Line A.

b. Graph a function h(x) that shifts the graph of f(x) 2 units up as Line B.

c. Write an equation for your new function h(x), in terms of f(x).

6. a. Graph g(x) = –2x + 4 as Line A.

b. Graph g(–x) as Line B.

c. What is different about these two graphs? What is the same?

7. Let g(x) = –2x + 4.

Find k such that the graph of g(kx) has slope 8.

k =

8. Let g(x) = –2x + 4.

Describe the relationships between the functions g(x), g(x) + k, kg(x), g(x + k), and g(kx).

9. For each formula determine whether a shift up, a shift down, a shift left or a shift right would be the result.

f(x) + k, when k < 0

f(x + k), when k > 0

f(x) + k, when k > 0

f(x + k), when k < 0

Challenge Problem

10. Suppose h(x) = 3x + 4 and i(x) = 4x – 5.

a. Find a and b such that h(ax + b) = i(x).

b. Find c and d such that i(cx + d) = h(x).



EXERCISES

EXERCISES

1. Suppose f(x) = –|4x – 1|. What is the value of f(–2)?

f(–2) =

2. What is the equation represented by this graph?

–6–8 –2–4 2 4 6 8

–4

–2

2

4

6

y

x

A f(x) = 2|x + 1|

B f(x) = |2x + 1|

C f(x) = |x – 1|

D f(x) = 2|x – 1|

LESSON 8: ABSOLUTE VALUE



EXERCISESLESSON 8: ABSOLUTE VALUE

3. What is the equation represented by this graph?

–6–8–10–12 –2–4 2 4

–4

–2

–6

2

4

y

x

A f(x) = |x – 5|

B f(x) = |x + 5|

C f(x) = –|x + 5|

D f(x) = –|x – 5|

4. a. Predict the relationship between the graph of f(x) = |x| and the graph of g(x) = |x| + 3?

b. Check your prediction by drawing each graph.

5. a. Predict the relationship between the graph of of f(x) = |x| and the graph of g(x) = –|x – 5|.


6. a. Predict the relationship between the graph of f(x) = |x| and the graph of g(x)= |2x|.


7. a. Predict the relationship between the graph of f(x) = |x| and the graph of

g x x( ) –= 12

.




EXERCISESLESSON 8: ABSOLUTE VALUE

8. An absolute value function has the slope ±2. Find the formula, given a minimum of (–2, 1).

9. An absolute value function has the slope ±2. Find the formula, given a maximum of (3, 0).

10. An absolute value function has the slope ±2. Find the formula, given a maximum of (–2, –2).

Challenge Problem

11. Suppose you start at one goal line of a football field, and you begin walking toward the other goal line. Write an equation relating x, the number of yards you have walked, and y, the yard line you are standing on.



EXERCISES

EXERCISES

1. What are the formulas, including domains, of this piecewise graph?

–6–8–10–12 –2–4 2 4

–4

–2

–6

2

4

y

x

A f(x) = x + 3 for x ≥ –3 –x – 3 for x < –3

B f(x) = x + 3 for x ≤ –3 –x – 3 for x > –3

C f(x) = x + 3 for x ≤ 3 –x – 3 for x > 3

D f(x) = x + 3 for x ≤ –3 –x – 3 for x < –3

2. What is the formula of the graphs combined?

A f(x) = –|x + 3|

B f(x) = |–x + 3|

C f(x) = –|–x + 3|

D f(x) = |–x – 3|–6–8–10–12 –2–4 2 4

–4

–2

–6

2

4

y

x

LESSON 9: PIECEWISE FUNCTIONS



EXERCISES

3. For each graph decide whether the graph is continuous or discrete.

Graph A: Graph B:

Graph C: Graph D:

–5 5 10 x

y

–5

5

–5 5 x

y

–5

5

–10 –5 5 10

–10

–5

5

10

y

x5 10

5

y

x




EXERCISES

4. For each graph decide whether the graph is a function or not a function.

Graph A: Graph B:

Graph C: Graph D:

–5 5 10 x

y

–5

5

–5 5 x

y

–5

5

–10 –5 5 10

–10

–5

5

10

y

x5 10

5

y

x




EXERCISESLESSON 9: PIECEWISE FUNCTIONS

5. Which of these statements are true about piecewise functions? There may be more than one true statement.

A Each piecewise function is defined by a single expression.

B A piecewise function can be continuous.

C A piecewise function can be discrete.

D A piecewise function has a constant slope for its entire graph.

E A piecewise function must pass the vertical line test.

6. Write the formulas, including domains, of the piecewise graphs.

–5 5 10 x

y

–5

5


–5 5 x

y

–5

5



EXERCISESLESSON 9: PIECEWISE FUNCTIONS


5 10

5

y

x

9. Graph this piecewise function.

f(x) = 1 for x < 1 2 for 1 ≤ x < 2 4 for x ≥ 2


f(x) = 1 for x < 1 4x – 3 for 1 ≤ x ≤ 2 5 for x > 2


f(x) = x + 1 if the greatest integer less than or equal to x is even x – 1 if the greatest integer less than or equal to x is odd

Challenge Problem

12. Describe the piecewise function that has a graph represented by y = –3|x – 10| + 4.



EXERCISESLESSON 10: LINEAR SEQUENCES

EXERCISES

1. The explicit rule for a linear (arithmetic) sequence is defined as t(n) = 6n – 16, where n is the term number starting with 1 and t(n) is the nth term.

Write the first 6 terms of this sequence.

2. Which of the following statements are true about a recursive rule for a sequence? There may be more than one true statement.

A A recursive rule uses the term number to determine the value of a specific term.

B The first term of the linear sequence is always 0.

C The rule and the initial term are needed to determine the sequence.

D The term number is a variable in a recursive rule.

E The initial term is a variable in a recursive rule.

3. Which of the following statements are true about linear sequences? There may be more than one true statement.

A The recursive rule is similar to the slope of a linear function.

B The first term of the sequence is similar to the y-intercept of a linear function.

C The domain of a linear sequence includes positive and negative integers.

D The graph of a linear sequence is continuous.

E The explicit rule depends on the previous term in the sequence.

4. What is the recursive rule for this sequence?

–50, –25, 0, 25, …

A t1 = –50, tn = tn – 1 – 50 for n > 1

B t1 = –50, tn = tn – 1 + 50 for n > 1

C t1 = –50, tn = tn – 1 – 25 for n > 1

D t1 = –50, tn = tn – 1 + 25 for n > 1



EXERCISESLESSON 10: LINEAR SEQUENCES

5. What is the explicit rule for this sequence?

–50, –25, 0, 25, …

A t(n) = 25n – 75

B t(n) = 25n – 50

C t(n)= 2n – 50

D t n n( ) = −12

50

6. The explicit rule for a linear (arithmetic) sequence is defined t(n) = 6n – 16, where n is the term number and t(n) is the nth term.

Find a recursive rule for this sequence.

7. The explicit rule for a linear (arithmetic) sequence is defined as t(n) = 6n – 16, where n is the term number and t(n) is the nth term.

Draw a graph of this sequence, where x is the term number and y the term value.

8. The explicit rule for a linear (arithmetic) sequence is defined as t(n) = 6n – 16, where n is the term number and t(n) is the nth term.

How is the graph of t(n) different from that of f(x) = 6x – 16? How is it similar? Explain.

9. Give the recursive and explicit rules for this sequence.

345, 347, 349, 351, …


12, 7, 2, –3, …


–1, –9, –17, –25, …

Challenge Problem

12. The 11th term of a linear sequence is 39, and the 25th term is 123. Find the explicit and recursive rules for this sequence.



EXERCISESLESSON 11: PUTTING IT TOGETHER

EXERCISES

1. Read your Self Check and think about your work in this unit.

Write three things you have learned during the unit.

Share your work with a classmate. Does your classmate understand what you wrote?

2. Sarah wrote this summary about linear relationships, but she did not finish her work. Read Sarah’s summary and finish the possible cases numbered 4–7.

There are four different formulas that show the relationshipbetween m as the slope, a as the x-intercept, and b as the y-intercept.

Formulas:

Function notationFormula to show -interc

: ( )f xx eept:

Formula to show x-intercept and -

y m x a

y

= −( )

iintercept:

Slope-intercept form:

xa

yb

y mx

+ =

= +

1

bb

m y yx x

m ba

Slope:

Slope:

=−

−

=−

2 1

2 1

a

b

m = change in ychange in x

m x= + bb

(continues)




(continued)

1. The slope, m , & y-intercepts (0, b) are knownn.

Find the x-intercept or a = bm

2. A

m ba

=−

− .

ppoint (p, q) and the slope, m, are known. Substtitute m and write an equation y = mx + bsubsstitute p and q for x and y. Solve for b.

3. x-iintercept (a, 0) & slope m

Substitute in m b=−aa

. Solve for b.

4. (p, q) & (r, s)

5. (pp, q) & y-intercept (0, b )

6. x-intercept (a, 0) & (r, s)

x-intercept (a, 0) & y-intercept (07. ,, b)

Here are possible cases of information given:

3. Sarah did not include in her summary what happens when linear functions or absolute value functions are transformed.

Write a summary including a graph of the effects on the graph of f(x) when it is linear and when it is an absolute value function:

f(x) + k, when k < 0

f(x + k), when k > 0

f(x) + k, when k > 0

f(x + k), when k < 0

kf(x), when k > 0

kf(x), when k < 0

Inverse of f(x)




4. Review the notes you took during the lessons about linear functions. Add any additional ideas you have about functions—including arithmetic sequences—to your notes.

5. Complete any exercises from this unit you have not finished.

Documents

LESSON 1: LINES AND LINEAR FUNCTIONS EXERCISESrusdmath.weebly.com/uploads/1/1/1/5/11156667/g9_u2_-_exercises_onl… · 11. Lian is taking guitar lessons with a private instructor