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Unit 9: Functions

Name: ___________________

Teacher: ______________

Period: _______

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Aim: SWBAT define domain, range, and function and determine functions given ordered pairs

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HW Functions: Determine whether the following relations are functions:

Example 1: Example 2: {(-2,0.5), (0, 2.5), (4,6.5), (5, 2.5)} {(6, 5), (4,3), (6,4), (5,8)} Example 3: Example 4: {(4.2,1.5), (5,2.2), (7,4.8), (4.2,0)} {(-1,1), (-2,4), (2, 4), (1,1)} Example 5: Example 6: {(-2,0.5), (0, 2.5), (4,6.5), (5, 2.5)} {(6, 5), (4,3), (6,4), (5,8)}

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State whether the following relations are functions. EXPLAIN YOUR ANSWER!

7) (-2, 4), (2, 4) (4, 2) (-2, -4) ______________________________________________

______________________________________________________________________

8) (3, -2), (6, 1) (-3, 5) (4, 1) ______________________________________________

______________________________________________________________________

9)

_____________________________________________________________________

______________________________________________________________________

10) (0, 2), (1, 4), (2, 6) (3, 8)

_____________________________________________________________________

______________________________________________________________________

11)

_____________________________________________________________________

______________________________________________________________________

Input -4 0 4 8

Output 8 0 8 32

Input 9 9 25 25

Output 3 -3 5 -5

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“Do Now” Which of the following sets of ordered pairs could represent a function? Why?

a) {(0,0),(1,1), (2,2), (3,3), (4,4)}

b) {(0,8),(1,6),(2,4), (3,2),(4,0)}

c) {(3,0), (3,1), (3,2), (3,3), (3,4)}

Aim: SWBAT determine functions from a graph, graph linear functions, and use function notation Vertical Line Test: _______________________________________________________ _______________________________________________________________________

Which of the following represent functions?

Evaluating Functions: y= -3x + 1 f(x) = -3x + 1 (read as “f of x”) f(0) = f(-1) = f(30) = The domain of f(x) = -2x+5 is {1,2,3,4} what is the range? _________________ The domain of f(x) = x2 is {1,2,3,4} what is the range? ___________________

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x (x, y) |x| + 3 -x2 (x, y) x

x (x, y) x2 - 2x + 3 x |x – 1| y

Graph each of these relations: Are they functions? Are the functions Linear ? 3 xy += 2xy −=

Y = |x – 1| y = x2 – 2x + 3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

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x x2 x -|x| + 2 (x, y) (x, y)

HW: Which of the following represent functions? Are the functions Linear ?

Graph each of these relations: Are they functions? Are the functions Linear ?

y = -|x| + 2 y = x2

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

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x (x, y) x

x

|x| - 4

|x + 4| x2 - 1

-x2 + 1

x (x, y)

(x, y)

(x, y)

Y = |x + 4|

y = x2 - 1

4- xy = y = -x2 + 1

-3

-2

-1

0

1

2

3

-7

-6

-5

-4

-3

-2

-1

-6

-4

-2

0

2

4

6

-3

-2

-1

0

1

2

3

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Do Now: For questions 1-3, state the domain and range. Then state if the relation is a function. 1. 2.

3. (3, 4) (4, 6) (2, 1)

4. Given f(7) = 13, what was the input, what is the output? Aim: SWBAT determine whether a function is a direct variation

Direct Variation:

List TWO criteria for items that are directly proportional:

• •

Equation for Direct Variation:

 y=Kx  m=  b=

The graph is always a ___________ equation.

Graph of direct variation must always pass through ______________ .

The constant of variation (unit rate) is always represented as the ________ of the line.

Graph the following equations: 1) y = 3x 2) 2y = -4x

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The graph of a direct variation equation is shown.

A) Write the direct equation B) Find the value of y when x = 20 3) 4)

Equation: _______________ Equation: _______________

If x = 20, y = _________. If x = 20, y = _________.

Find the value of x

y for each ordered pair. Then, tell whether each relationship is a direct

variation. If so write the direct variation equation.

5)

____________________________________

x 1 3 5

y 12 36 60

x

y

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6)

____________________________________

7) The value of y varies directly with x, and y = 35 when x = -5, find y when x = -20.

8) The value of y varies directly with x, and y = 48 when x = 4, find x when y = 36.

9) Which graph represents a direct variation?

A) B) C)

x 30 45 60

y 40 60 90

x

y

13

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“Do Now” In each case: a)What is the constant of variation? b)Write an equation for y in terms of x c)What is the slope of the line? 1) The perimeter of a square (y) is 12 cm. when the length of a side of the square (x) is 3 cm.   2) A printer can type 160 characters (y) in 10 seconds (x) 3) The length of a photograph (y) is 12 cm when the length of the negative from which it is developed (x) is 1.2 cm. 4) Three pounds of meat (y) will serve 15 people (x) Aim: SWAT determine direct variation from word problems and graphs

Direct Variation : List TWO criteria for items that are directly proportional:

• •

Equation for Direct Variation:

 y=Kx  m=  b=

The graph is always a ___________ equation. Graph of direct variation must always pass through ______________ . The constant of variation (unit rate) is always represented as the ________ of the line.

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Direct Variation--A relationship between two quantities (variables) so that the ratio is constant. We say that one quantity (variable) varies directly as the other. Or that one variable is directly proportional to the other. Constant of Variation— Ratio (slope) 1) To make lemonade from frozen concentrate Savannah uses three cans of water for every can of lemonade concentrate. If Savannah wishes to make 4 cans of concentrate, how many cans of water must she use?

What is the ratio cans of water to cans of lemonade concentrate?______________________

Is the ratio constant? ______________________

If there is no lemonade concentrate to mix how much water is needed? __________________ In the above example, does the number of cans of water vary directly as the number of cans of lemonade concentrate used? ___________ If so, what is the constant of variation?__________

Solve for y:

Write the direct variation equation: _____________

Graph the relationship.

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2) The amount of flour needed to make a white sauce varies directly as the amount of milk used. To make white sauce, a chef used 2 cups of flour and 8 cups of milk. Write an equation and draw the graph of the relationship.

3) If four tickets to a show cost $17.60, if sold at the same rate what is the cost of seven tickets? Write an equation for this relationship.

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HW:

Find the value of x

y for each ordered pair. Then, tell whether each relationship is a direct

variation. If so write the direct variation equation.

1)

____________________________________

2)

____________________________________

3) The value of y varies directly with x, and y = 520 when x = 13, find y when x = 9.

4) The value of y varies directly with x, and y = 10 when x = 8, find y when x = 12.

5) Nick scores an average of 7 foul shots in every 10 attempts. At the same rate, how many

shots would he score in 200 attempts?

6) If 3 pounds of apples cost $0.89, what is the cost of 15 pounds of apples at the same rate?

x 1 2 3

y 20 40 60

x

y

x 2 3 4

y -6 -9 -12

x

y

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Interpreting Data from Tables and Graphs

1) The students in an eighth-grade class had a dance. They spent $500 for a local band. The

equation y = 4x – 500 can be used to find the total profit, y, if the students sold x tickets to

the dance.

What does the 4 represent in the equation?

A. the price per ticket

B. the cost of the band

C. the number of tickets sold

D. the profit made from selling x tickets

2) Elena and Kristen started new jobs at the same time. The table below shows their annual

salaries for the first 4 years.

Elena’s salary continued to increase by the same amount each year,

and Kristen’s salary continued to increase by the same amount each year. Which of the following

statements is true for year 6?

A. Elena’s salary was $30,000.

B. Kristen’s salary was $26,000.

C. Elena’s salary was $500 more than Kristen’s salary.

D. Kristen’s salary was $500 more than Elena’s salary.

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3) The relationship between the perimeter and side length of a regular hexagon is shown on

the graph below.

a) Is the relationship between the perimeter of a regular hexagon and the length of its side

proportional or linear? How do you know?

b) What happens to the perimeter of a regular hexagon as its side length increases by 1?

A. The perimeter increases by 1.

B. The perimeter increases by 2.

C. The perimeter increases by 3.

D. The perimeter increases by 6.

c) As the length increases by 1, the perimeter increases by the same amount each time. What

does this constant increase represent graphically?

a) The y-intercept of the line. b) The slope of the line. c) The x-intercept of the line. d) The length of the line.

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4) Marisa drank one cup of milk and ate x small vanilla cookies for a snack. The cup

of milk had 120 calories, and each cookie had 12 calories. Write an equation to

represent the total number of calories, y, in Marisa’s snack.

a. What is the y-intercept of the line represented by this equation?

b. Explain what the y-intercept tells us about Marisa’s snack.

c. What is the slope of the line represented by this equation?

d. Explain what the slope tells us about Marisa’s snack.

e. If Marisa eats 9 small vanilla cookies, what is the total number of calories in her snack? Show or

explain how you got your answer.

Graph the relationship below:

Number of Cookies

Num

ber of C

alories

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5) The motion graph below shows a woman walking away from a starting point.

a) Describe her motion between

0 and 2 seconds.

b) Calculate the woman’s speed, in meters per second, for each of the following time intervals:

2 to 4 seconds: ___________________________

4 to 6 seconds: ____________________________

6 to 8 seconds: ____________________________

6) Does the equation y = 3/x define a linear function? Explain.

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AIM: SWBAT write function rules. To write a function rule that relates input ( ) and output ( ) try to find an equation of the form

y = ax + b

The value of a = change in output y

change in input x =

Δx

Δy

Steps for finding the function rule

Use the form y = ax + b

I. Find the value of “a” a = Δx

Δy

II. Substitute the “a” value into y = ax + b

III. Choose a given ordered pair (x,y) and substitute into y = ax + b

IV. Solve equation for “b”

V. Final Step in y = ax + b replace the “a” and “b” values to find the FUNCTION RULE

Lets try: Given:

How did “x” values change? Increase by ______

How did “y” values change? _________________

Use y = ax + b Function Rule is :________________

I. Find “a”

a = Δx

Δy =

II. Plug value of “a” into the form y = ax + b

III. Choose a given ordered pair and sub into y = ax + b ( , )

IV. Solve for “b”

Input x -4 -2 0 2 4

Output y 1 3 5 7 9

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Classwork:

Given the following functions find each function rule. Show ALL work step-by-step.

1) (0, 3), (1, 5), (2, 7)

2) (3, 3), (5, 7), (-4, -11)

3)

x -2 0 2 4

y -3 -2 -1 0

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Homework – Writing Function Rules

Write a function rule that relates x and y.

1)

x -2 -1 0 1

y 5 6 7 8

_______________________________________________________________________

2)

x -3 -2 -1 0

y 8 5 2 -1

_______________________________________________________________________

3)

x y

0 -3

2 -2

4 -1

6 0

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DO NOW: Answer each question.

1) y varies directly as x. If y = 40 when x = 50 write the direct variation equation.

K = _______ y = _____________

Use the equation that you wrote above to find the value of y when x is 7.4

2) The relation in the table below names a function where x is the input and y is

the output.

a) List the members of the domain: ________________

b) List the members of the range: ________________

c) Write an equation that could be used to represent this function. ________________

(Show work)

d) Is this a linear function? _____________

3) Given f(x) = -2x + 7. If the domain is {-3, 0, 3} find the range. (Show all work.)

x y

-7 0

-2 5

3 10

8 15

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4) An electrician charges $90 for a house call, plus $60 for each hour of work. Write an

equation that shows how the total cost of the electrician’s work, y, depends on the number

of hours, x, the electrician works. What is the rate of change and initial value?

How much did the electrician charge for a 4 hour house call?

Equation ____________________

Rate of Change _______________

Initial Value _______________

Cost of a 4 hour house call _______________

5)

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Name ________________________________________ Math 8 Function REVIEW

1. Which equations are linear functions?

A) y = x2 + 4 B) x + y = 4 C) y = 4x + 4 D) 8x + 8y = 4

2. Which equations are non-linear equations?

A) -2x – 4y = 12 B) x2 + y =3 C) 2xy + 3 = 12 D) x2 = y

3. Which of the following relations are functions? Why?

A) (2,-4), (-3,2), (2,3), (-2,3) B) (5,-4), (-4,5), (-6,4), (4,-3)

C) (8,-5), (-5,8), (5,-9),(-8,5) D) (-7,6), (5,-7), (7,9), (10,3)

4. Which table represents a function? Draw an arrow diagram for each.

5. Identify the function rule that relates the x and the y for the following function:

x 0 1 2 3

y 1 4 7 10

What is the independent variable? What is the dependent variable?

What is the domain? What is the range?

What are the inputs? What are the outputs?

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6. Which equations represent a direct variation? Why?

A) y = x B) y = x + 2 C) y = 3x – 1 D) y = 2x

7. What is the rate of change represented by the table below?

8. For each graph below, is it a function? Why or why not? Is it linear or non-linear?

x -1 0 1 2 3

y 1 3 5 7 9

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9. The table represents the linear function y = -3/2 x + 9. What is the missing output value?

What is the independent variable?

10. Sam went kayaking to a nearby island. The graph below

shows his trip.

During which part of the trip was Sam paddling the fastest?

During which part of the trip did Sam stop on an island for

lunch?

11. Is the graph below a function? Is it a linear function? Is it a direct variation?

What is the rate of change?

12. Given the domain of the function f(x) = 2x – 5 to be {-1, 0, 1}

Input Output

-8 21

-2 12

0 9

6 ?

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find the range.

13. Given the notation f(5) = 3x + 2, what is the input? What is the output?

14. The relation represented in this table names a function where x is the

input and y is the output:

a) List the members of the domain: ____________

b) List the members of the range: ____________

c) Write an equation that could be used to represent this function.

d) Is this a linear function? _____________

15. Y varies directly as x. If y = 10 when x = 20 write the direct variation equation.

Use the equation that you wrote to find the value of y when x is 8.6

16. A printer can type 160 characters (y) in 10 seconds (x). At that same rate, how many characters can

it print in a minute?

x y

-2 12

-1 7

0 2

1 -3

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