Name __________________________

Math 3 – Unit 1: Functions & Lines

Day

Topic

Assignment

1

Relations, Functions, Domain, Range

CW: pages 2,3

Page 3,4 #1-29

2

Domain, Range, Graphing Lines

CW: page 5, p. 6 #1-8, page 7 all

Page 6 #9-16, page 8

3

Graphing lines with translations

Writing Equations CW: p. 9, 10

Bottom of page 4

Page 11

4

Quiz #1

Absolute Graphs

CW: p. 12

Page 13

5

Linear Equations in Context

CW: page 14

Page 15

6

Group Quiz

Page 16

7

Unit 1 Test

Unit 2 packet page 2

CCM 3 Unit 1 Review Relations and Functions

Relation: a set of pairs of input and output values

D

I

X

R

O

Y

Function: A relation in which each element of the domain is paired with exactly one element in the range

Ways to Represent

Function Example

NOT a function

Numerically

Algebraically

Graphically

Verbally

Determine if the following are functions.

1) 2) T(s), where t is teacher and s is subject3) 4)

5) {(-1, 5), (2, 5), (3, 5) (8, 5)}6) P(b), P is person and b is birthday7) B(p), where b is birthday and p is person

8) 9) {(5, 1), (5, 2), (5, 3)} 10) 11) y = |x – 2| 12) y=

13) A national park contains foxes that prey on rabbits. The table below gives the two populations, F and R, over a 6-month period, where t=0 means January 1, t=1 means February 1, and so on.

t, month

0

1

2

3

4

5

R, rabbits

1000

750

567

500

567

750

F, foxes

150

143

125

100

75

57

a) Is F a function of t? Explain and discuss with a partner.

b) Is R a function of t? Explain and discuss with a partner.

c) Is F a function of R? Explain and discuss with a partner. d) Is R a function of F? Explain and discuss with a partner.

Finding a Function Value:

Numerically:

1. f(1) = 2. f(2)=3. g(4)=4. 2f(4)+1 5. f(4+1)

6. 4g(3(2))7. If f(x) = 2, what is x? 8. If g(x) = 0, what is x?

Graphically:

f(1) = f(3) =

If f(x) = 4, what is x?

Verbally:

s(t) represents the speed of a car in mph at time t in hours.

Explain what the following mean using words.

s(3) = 60 s(7) = 0

Algebraically:

If f(x) = 3 – x², g(x) = x² + 1, and h(x) = 3x + 2 find:

1) h()2) f(2) 3) g(-1)4) g(3x)

5) f(-3)6) g(4)7) 2h(5)8) h(2(5))

9) f(x + 2)10) f(x) + 211) g(x – 3)12) h(x + 4)

HW #1

For 1-13, are the following relations functions? Explain why or why not.

1. (2, 6) (-3, 6) (4, 9) 2. (-2, 4) (-2, 6) (0, 3)3. (1, 3) (2, 3) (3, 3)4. (-2, 5) (-1, -1) (3, 7) 5) (0, 1) (2, 4) (4, 1) (0, 5)

6. h(t) where h is the height of a ball thrown in the air after t time7. S(h), S is shoe size, h is height.

8. Y(m), the Y=year born m=month born 9. t(h) where t is the time elapsed and h is height of a ball thrown in the air

10. 11. 12. 13.

x

f(x)

g(x)

Using the table at the left find:

14. f(0) 15. g(2)

16. f(2) 17. g(0)

18. 2f(1) 19. f(2(1))

20. If g(x) = -1, what is x? 21. f(g(0))

22. if f(x) = 1, what is x?

-1

1

4

0

7

2

1

2

-1

2

8

4

For the following, use the functions f(x) = x + 5 and g(x) = x² + 3. Find:

23. f(4) 24. f(-2) 25. f(6x + 4)

26. f(6n)27. g(4)28. 3g(-2) 29. g(x + 4)

Domain and Range Express using units if possible and appropriate values

1. L(t) represents the length of a burning candle after t minutes.

Domain: Range:

2. h(t) is the height after t seconds of a yo-yo moving up and down rhythmically.

Domain: Range:

3. g(d) is the number of gallons of gas used after traveling d miles on flat land.

Domain: Range:

4. (1, 2) (3, 5) (-2, π)

Domain: Range:

Interval Notation:  (description)

(diagram)

Open Interval:   (1, 5)  is interpreted as 1 < x < 5  where the endpoints are NOT included.(While this notation resembles an ordered pair, in this context it refers to the interval upon which you are working.)

Closed Interval:  [1, 5]  is interpreted as 1 < x < 5  where the endpoints are included.

Half-Open Interval:  (1, 5]  is interpreted as 1 < x < 5 where 1 is not included, but 5 is included.

Half-Open Interval:  [1, 5) is interpreted as 1 < x < 5 where 1 is included, but 5 is not included.

Non-ending Interval:   is interpreted as x > 1 where 1 is not included and infinity is always expressed as being "open" (not included).

Non-ending Interval:   is interpreted as x < -5 where 5 is included and again, infinity is always expressed as being "open" (not included).

Write the following descriptions using interval notation. Draw a number line if you need to.

1. 2. 3. 4.

5. 6. 7.

8. 9. All real numbers 10. All real numbers except for 5

11. All real numbers except for 2 and -212. All even integers

All answers must be in interval notation, if possible.

Find the domain of the following relations and determine if it is a function.

1.

D:

Function?

2.

D:

Function?

3.

D:

Function?

4.

D:

Function?

Find the range of the following relations.

5.

Range:

6.

Range:

7.

Range:

8.

Range:

HW: Find the domain and range of the following relations and determine if it is a function.

9)10)11)12)

D:

R:

Function?

D:

R:

Function?

D:

R:

Function?

D:

R:

Function?

13)14)15)16)

D:

R:

Function?

D:

R:

Function?

D:

R:

Function?

D:

R:

Function?

Part I: Graphing LinesSlope-intercept form of a line:

To graph:

1) Graph your y-intercept ( ) on the y-axis.

2) From that point, use your slope to graph the second point.

1. y =

m= b=

Graph #1

y

x

6. y = -2

m= b=

Graph #6

y

x

11. y = 2x

m= b=

Graph # 13

2. y = -3x + 1

m= b=

7. x = 4

m= b=

12. f(x) = -5

m= b=

3. y =

m= b=

8. f(x) = 3x – 1

m= b=

13. x = -3

m= b=

4. f(x) = x + ½

m= b=

9. y =

m= b=

14. y =

m= b=

5. y = x

m= b=

10. y = x – 1

m= b=

15. f(x) = x

m= b=

To graph in standard form:

1. Find the x-intercept, which is when ____ equals zero. This gives you the point (____, _____)

2. Find the y-intercept, which is when ____ equals zero. This gives you the point (____, _____)

3. Alternative: get into slope-intercept form (Solve for y)

State the x and y intercept and graph.

16) 6x + 3y = 1217) 2x – 5y = 1018) 5x – y = 219) 2x + 3y = 5

y

x

y

x

y

x

y

x

CCM 3 Unit 1 Homework

Find the slope and y-intercept and graph.

1) y = 2x2) 2x – y = 33) x = -44) 4x + 5y = 20

y

x

y

x

5) y = -4x + ½ 6) y = 37) y = 3x – 28) y = -x + 4

y

x

y

x

9) 2x – 5y = -1010) y = x11) y = 12) y = -x + 7

y

x

y

x

13) 4x – 3y = -614) 3x – 2y = 915) y = -3x – 1

y

x

y

x

Part II: Translations of Linear Functions NEED 3 COLORS, ONE FOR EACH GRAPH

1. A. Graph y = 2x. Plot at least 4 points on the line.

B. Translate the graph 4 units to the right by moving each point 4 units to the right.

Write the equation of the translated line in slope-intercept form.

Factor out the common factor in the translated equation.

What are the coordinates of the point to which the point (0, 0) is translated?

C. Translate the graph 3 units to the left by moving each point 3 units to the left.

Write the equation of the translated line in slope-intercept form.

Factor out the common factor in the translated equation.

What are the coordinates of the point to which the point (0, 0) is translated?

D. In each factored equation, what does the 2 tell you about the line?

E. What does the number inside the parenthesis tell you about the line?

F. What does the sign of the number inside the parentheses tell you about the line?

2. A. Graph y = x. Plot at least four points on the line.

B. Given y = (x + 6)

the slope would be ____

the graph would translate __________.

the point (0, 0) would move to ______. Graph this line.

C. Given y = (x – 3)

the slope would be ____

the graph would translate __________.

the point (0, 0) would move to ______. Graph this line..

D. A line with a slope of ½ that goes through the point (5, 0) would have the equation:

E. A line with a slope of ½ that goes through the point (-4, 0) would have the equation:

3. A. Graph y = -3(x – 4). Plot at least four points on the line.

B. Translate the graph in part (A) up 5 units by moving each point 5 units up.

The (4, ___) is on this new line.

The equation of this line can be written as y = -3(x – 4) + 5

Show that if x = 4 then y = 5.

C. Translate the graph in part (A) down 2 units by moving each point 2 units down.

The (4, ___) is on this new line.

The equation of this line can be written as y = -3(x – 4) – 2

Show that if x = 4 then y = -2.

D. Given y = 3(x + 2) + 4

The slope would be _______

The point (___ , ____) would be on the line. Graph it.

D. Given y = 3(x + 2) – 5

The slope would be _______

The point (___ , ____) would be on the line. Graph it.

POINT-SLOPE FORM OF A LINE

y = m(x – h) + k

4A. Given y = (x – 2) + 3

The slope would be ____

A point on the graph would be _______.

Graph this line.

B. Given y = 2(x + 1) – 3

The slope would be ____

A point on the graph would be _______.

Graph this line.

C. A line that has a slope of 4 and passes through the point (2, 1) would have the equation:

D. A line that has a slope of 4 and passes through the point (-3, -5) would have the equation:

5. Without graphing, list the slope and name a point on the following lines.

a) y = 3(x – 2) + 4 b) y = ½ (x + 3) – 5 c) y = 2x + 6

6. Write an equation in point-slope form for the graph described. Then put it in slope-intercept form.

a) slope is 2 through the point (4, -3)b) slope is -1 through the point (-2, -7) c) slope is ½ through the point (0, 5)

7. Write an equation in point-slope form for the graph described.

a) || to y = 3x – 2 through (7, 4)b) to y = -2x + 8 through (-17, 80) c) to y = +5 through (10, -π)

8. Write an equation in point-slope form for the graph described.

a) slope of -7 through (0, 2) b) through (2, -5) and (8, 10)c) through (-2, -4) and (8, 3)

Practice

1) Without graphing, name the slope and a point on the following lines.

a) y = 2(x + 4) – 3b) y = -(x + 2) + 1c) f(x) = 2(x – 1) – πd) e)

f) y = -5x + 2g) y = -5(x + 12)h) y = i) f(x) = j) y = 1 k) x = -8

2) Write an equation in point-slope form for the graph described.

a) translates y = 3x 7 right and 2 upb) through (2, 8) and (5, -3)c) || to y = 4x – 2 through (9, -3)

d) to y = 2x + 1 through (-5, 8)e) slope is 0 through the point (-3, -10) f) through (-8, -7) and (-3, 8)

g) || to y = through (2, -9)h) translates y = x 10 left and 1 downi) to f(x) = through (0, -6)

j) through (-1, -3) and (5, -17) k) through (2, -5) and (2, -11)l) through (-4, 7) and (-10, -2)

Answers NOT in order

y = -10

m = -5,

pt (-12, 0)

m = -5, pt (0, 2)

y =

m = -3; pt

y =

y =

m = -1;

pt (-2, 1)

m= ; pt (π, -2)

y =

y =

m = und, pt (-8, 0)

m = 0, pt (0, 1)

y = (x + 10) – 1

y =

m = ; pt (0, -23)

m = ; pt (17, 0)

x = 2

m = 2; pt (-4, -3)

m = 2;

pt (1, -π)

y = 3(x – 7) + 2

y = 3(x + 8) – 7

y = 4(x – 9) – 3

Absolute Value Graphs and Transformations:

x y

Graph y = |x|Graph y = -|x|

x y

y = a|x – h| + k translates y =a|x| to vertex at ( , )

Identify the vertex. Graph. Identify the domain and range.

1) y = |x – 3| + 1 2) y = - | x + 2| - 4 3) y = | x + 5| + 3 4) y = 3 | x – 1| - 5

Vertex: Vertex: Vertex: Vertex:

D: D: D: D:

R: R: R: R:

5) y = ½ |x + 2| 6) y = - |x – 3| 7) y = |x| - 1 8) y = -2|x| + 39) y = |x + 4| + 2

Vertex: Vertex: Vertex: Vertex: Vertex:

Transformation: Trans.: Trans.:Trans.:Trans.:

D: D: D: D:D:

R: R: R: R:R:

Write an equation given the transformations.

10) y = |x| vertical stretch by a factor of 3, left 5 , down 2 11) y = |x|; reflected across x-axis, right 2 , down 4

12) y = |x| vertical compression by a factor of 1/4 , left 8, up 713) y = |x|; reflected across the x-axis, up ½

14)15)16)) 17)

HW: Determine if the graph is a function, then find the domain and range.

1) 2) 3) 4)

x

f(x)

g(x)

5) find a) f(-1 + 1) b) 3(g(0)) c) If f(x) = 2, what is x?

d) if g(x) = 2, what is x? e) g(f(1))

-1

1

4

0

7

2

1

2

-1

2

8

4

Determine if the following are functions:

6) C(p) is the class for a student during period p. 7) D(t) is the distance from the starting line of a runner at time t

8) G(q) is the grade given the number of questions q right9) B(h) is the dog breed given the height h of a dog

10) Solve 10(x + 3) – (-9x – 4) = x – 5 + 3

Answers to 1-10 (not in order)

(-4, 4]

-2

0

1

4

6

7

N

Y

Y

Y

Graph the following and find the domain and range.

11) y = - |x + 2| - 3 12) y = -½ |x + 3| 13) y = 2|x| + 2 14) y = 3|x −2| − 5

D: D:D: D:

R: R: R: R:

For 18 – 21, write an equation given the transformations or the graphs below for 22-25.

15) y = |x|, up 4 units16) y = |x|, reflect over x-axis, down 2 units17) y = |x|, vert. stretch by 2, left 3 units

18) y = |x|, vertical compression by 1/6, right 8 units

Linear Equations in Context

Example 1: 1. Electric Avenue sells audio/video, computer, and entertainment products. The store offers 0% interest for 12 months on purchases made using an Electric Avenue store credit card. Emily purchased a television for $480 using an Electric Avenue store credit card. The chart below shows how Emily’s account balance changes as she makes monthly payments.

number of monthly payments

0

1

2

3

4

5

6

account balance ($)

480

460

440

420

400

380

360

a) Is the account balance a linear function of the number of monthly payments? Why?

_________________________b) What is the rate of change in the account balance as the number of monthly

payments increases? Be sure to include the units for the rate of change.

____________________________c). Write an algebraic rule.

d) Will Emily pay off the balance within 12 months? ______________________________

Example 2: The average lifespan of American women has been tracked, and the model for the data is

y = 0.2t + 73, where y= the life span and t = 0 corresponds to 1960. 

_________________________a) What is the slope and what does it represent in the context of the problem?

_____________________b) What is the y-intercept and what does it represent in the context of the problem?

____________________________________c) Use the model to predict the lifespan of a woman in 2015.

Example 3: Take a look at the graph.

This graph shows how John's savings account balance has changed overthe course of a year. We can see that he opened his accountwith $300 and by the end of the first month he had saved $100. By the endof the 12 month time span, John had $1500 in his savings account.

_________________a) About how much was John saving per month?

_______________________________________b). Write an algebraic rule.

4) The taxi fare for a 1 km trip within the limits of New York City is $1.70. A trip of 3 km costs $3.30. What is the fare for a 2 km trip?

The basic cost for a taxi ride is _____ and each kilometer costs ______ extra.

5) A mass of 40 g causes a wire spring to stretch to a length of 6.2 cm, and a mass of 60 g stretches the spring to a length of 6.7cm. What will be the length of the spring if a 90g mass is attached?

The original length of the spring is _______ and each _____ added adds ________ to the length.

_________________c) How much money will John have saved after 2 years?

Write a linear equation for each problem, fill in the blanks, then solve.

Using Linear Equations Practice

1. Creative Catering charges a basic fee of $100 to cater a banquet plus an additional charge of $2 per person invited to the banquet.

a) Define variables: The ________________ depends on _____________________.

Let y = Let x = b) Write an algebraic rule.

2. You are selling foot-long hot dogs for $2.50. You had to spend $150 to invest in a hot dog cart.

a) Define variables: The ________________ depends on _____________________.

Let y = Let x = b) Write an algebraic rule.

Write a linear equation for each problem, fill in the blanks, then solve.

3) At 15°C the length of a metal rod is 72.6 cm. At 95°C, its length is 72.8 cm. How long is the rod when its temperature is 75°?

Each degree in temperature adds _____ to the starting length of _______.

4) The toll for a telephone call of 2 minutes between two particular cities is $0.79 and the toll for a 5 minute call is $1.60. What is the toll for a call of 9 minutes?

The basic cost for a phone call is _____ and each ________ costs ______ extra.

5. A health club charges a yearly membership fee of $95 and members must pay $2.50 per hour to use its facilities.

a) Define variables: The ________________ depends on _____________________.

Let y = ____________________Let x = _________________b) Write an algebraic rule. ___________________

_____________c) How many hours did Paul use the club last year if his bill was $515?

6. The average lifespan of American men has been tracked, and the model for the data is y = 0.1t + 71.8, where y= the life span and t = 0 corresponds to 1990. 

______________________a) What is the slope and what does it represent in the context of the problem?

__________________b) What is the y-intercept and what does it represent in the context of the problem?

__________________c) Use the model to predict the lifespan of a man in 2015.

number of videos rented in a month

1

2

3

4

5

10

monthly bill ($)

18

21

24

27

30

45

7. To rent videos from the Videorama store you pay a monthly membership fee, and then a fee for each video you rent. The chart below shows how George’s monthly bill changes based on the number of videos he rents in a month.

_________a) Is the George’s monthly bill a linear function of the number of videos rented in a month? Why?

__________b) What is the monthly membership fee? _______________c) What is the charge per video?

___________d) Write an algebraic rule. ____e) How many videos did George rent if his monthly bill was 81?

Study GuideWRITE ON SEPARATE SHEET!

Are the following relations functions?

1) (3,4) (3,7) (3,8)2) (3,3) (4,4) (5,5)3) (2,3) (4,5) (6,7)4)5)6)

7) A(s), the artist given the song title8) A(n), the author given the novel

Given f(x) = -x² – x and g(x) = x² + 3, find the values. 9) f(4) 10) g(5) 11) g(-5) 12) 3f(-4) 13) g(x – 4)

x

f(x)

g(x)

Use the chart to the left.

14) Find f(3) 15) If f(x) = 2, what is x?

16) Find g(1) 17) If g(x) = 1, what is x?

1

3

0

2

-1

1

3

2

-2

Find the domain and range in interval notation.

18) 19) 20)

Graph on graph paper.

21) y = x 22) y = -2x + 323) y = 24) 2x + 3y = -1225) y =

26) y = 2 27) y = –(x – 4) – 228) x = -3½29) y = |x + 2|-130) x – 2y = 5

31) y = 32) 3x – y = 633) y = -3|x – 4|+2 34) y = ½ |x + 1| + 3

Give an equation in point-slope form for the line described.

35) through (3,-2) and (8, 4) 36) through (-2,0) and (-3, 8) 37) to y = through (8, 2)

38) slope = 0 through (3, -1) 39) translates y = 3x left 1 up 2 40) slope = undefined through (4, 2)

Give an equation in slope-intercept form for the line described.

41) translates y = -2x right 3, down 742) through (5, 2) and (8, -7) 43) || to y = through (-6, 10)

44) translates y = x left 7, up 5 45) to y = through (-12, 9)

Answers (in mixed order)

D

R

D [-5, -2] U (0, )

R:

D (-4, -1)

R (-1, 3]

D{1, 2, 3} R{-1, 2, 3}

y =

y =

x = 4

no, does not pass vertical line test

no, does not pass vertical line test

no, 3 has more than one output value

no, songs can be covered by more than one artist

y =

y =

y = -8(x + 2)

y = -3x + 17

y =

y = -2x – 1

y = -1

-36

-20

0

2

2

3

28

28

x² – 8x + 19

yes

yes

yes

yes, a novel has just one author

y = 3(x + 1) + 2

graphs are on key on my website!

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