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Unit 2 – Part 2 – Special Functions Name: ____________________________

Piecewise Functions - Graphing Shipping Rates

The following information comes from the United States Postal Service website, www.usps.com

You will use your calculator to generate 5 random package weights, which you will place on the axes on the front board.

To get your package weights:

In the scratchpad on your calculator, choose MENU 5: Probablility 4: Random 6: Seed Enter your cell phone number and hit enter MENU 5: Probablility 4: Random 1: Number Bump out of the parentheses, multiply by 8, and hit enter 5 times

Enter your 5 points into the class Desmos table:

Weight Cost

1. What do you notice about our graph?

2. What do you wonder?

3. What characteristics of this graph are similar or different from graphs you’ve seen before?

What about the context causes these similarities/differences?

4. Is this relation a function? If so, state the domain and range.

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Piecewise Functions Introduction 1) You have a summer job that pays time and a half overtime. That is, if you work more than 40 hours per week, your hourly wage for the extra hours is 1.5 times your normal hourly wage of $7.

a) How much will you make if you work 35 hours? 40 hours? 45 hours? b) Write an expression for wages earned if you work up to 40 hours. c) Write an expression for waged earned if you work over 40 hours. d) Let’s practice writing the piecewise function:

Number of Hours

worked

Money

earned

0

10

20

30

35

40

41

42

43

44

45

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2) You and your friends decide to go to Detroit for a Red Wings game. You find a parking garage with the following rates: $2 per hour for the first 3 hours and $8 maximum for over 3 hours.

a. Fill out the table b. Make a graph

c. How much would it cost to stay for 10 hours? When would the cost be exactly $2? Exactly $3? d. Let’s see if we can write the piecewise function: Practice Evaluating Functions: Evaluate the function for the given value of x.

2) 𝑓(𝑥) = {5𝑥 − 1, 𝑖𝑓 𝑥 < −2𝑥 − 9, 𝑖𝑓 𝑥 ≥ −2

a) f(-4) b) f(-2) c) f(0) d) f(5)

time cost time cost

0 2 hours

15 mins 2 hours 15 mins

30 mins 2 hours 30 mins

45 mins 2 hours 45 mins

1 hour 3 hours

1 hour 15 mins 4 hours

1 hour 30 mins 5 hours

1 hour 45 mins 6 hours

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3) ℎ(𝑥) = {2𝑥, 𝑖𝑓 𝑥 ≥ 1

−𝑥 + 3, 𝑖𝑓 𝑥 < 1 a) h(1) b) h(-10) c) h(6) d) h(0)

Piecewise Functions

Piecewise Function - _____________________________________________________________________

Example: 1. Evaluate the function for the given value of x.

𝑓(𝑥) = {3𝑥 − 1, 𝑖𝑓 𝑥 ≤ 42𝑥 + 7, 𝑖𝑓 𝑥 > 4

a. x = 10 b. x = 4 c. x = -2 d. x = 1

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2. Evaluate the function for the given value of x.

𝑓(𝑥) = {5𝑥 − 1, 𝑖𝑓 𝑥 ≤ 1−𝑥 + 4, 𝑖𝑓 𝑥 > 1

a. x = 1 b. x = 4 c. x = -2 d. x = 1

3

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Rules for graphing a piecewise function:

1. Graph each equation, one at a time, over its specified domain.

2. If or , you will use a ____________ circle for the endpoints.

If < or > , you will use a ____________ circle for the endpoints.

Example: Graph the functions.

1

4, 12

2 3, 1

x xf x

x x

2 5, 2

3 2, 2 1

2, 1

x x

f x x x

x

x y

x y

x y

x y

x y

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During a nine hour snowstorm it snows at a rate of 1 inch per hour for the first two hours, at a rate of 2 inches

per hour for the next 6 hours, and a rate of 1 inch per hour for the final hour.

a. Write and graph a piecewise function that gives the depth of the snow during the snowstorm.

b. How many inches of snow accumulated from the storm?

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Introduction to Absolute Value Graphs – Card Sort Examine all of the graphs, tables, and equations you have. 1. What do all of the graphs have in common?

___________________________________________________________________________________________

___________________________________________________________________________________________

2. What do all of the equations have in common?

___________________________________________________________________________________________

___________________________________________________________________________________________

3. What do all of the tables have in common?

___________________________________________________________________________________________

________________________________

___________________________________________________________________________________________

Sort all of the graphs according to some commonality among the graphs. Find at least three different ways to sort the graphs. You have been provided with space for 4 different groupings, but you may only need 2 or 3 spaces, depending on your commonality. Sort #1 We grouped the following graphs together (give the letter of the graphs you grouped together): Grouping 1: _________________________ Grouping 2: _________________________ Grouping 3: _________________________ Grouping 4: _________________________ We grouped the graphs together, because ______________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Sort #2 We grouped the following graphs together (give the letter of the graphs you grouped together): Grouping 1: _________________________ Grouping 2: _________________________ Grouping 3: _________________________ Grouping 4: _________________________ We grouped the graphs together, because ______________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________

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Sort #3 We grouped the following graphs together (give the letter of the graphs you grouped together): Grouping 1: _________________________ Grouping 2: _________________________ Grouping 3: _________________________ Grouping 4: _________________________ We grouped the graphs together, because ______________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________

Absolute Value Functions

Absolute Value: __________________________________________________________________________

Plug the following values into the function and then plot the points.

1. xy

x xy y ( x , y )

-2

-1

0

1

2

9

2. 23 xy

General form of an absolute value equation: khxay

Vertex: ( , ) Slope:

(Always go right first!)

Example: Graph the functions.

a. 32 xy b. 412 xy

a = _______ a = _______

( h , k ) = _________ ( h , k ) = _________

x 23 xy y ( x , y )

1

2

3

4

5

10

c. 32 xy d. 112

1 xy

a = _______ a = _______

( h , k ) = _________ ( h , k ) = _________

Example: Write an absolute value equation for the graph shown.

a. b.

a = _______ a = _______

( h , k ) = _________ ( h , k ) = _________

Equation: Equation:

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c. d.

a = _______ a = _______

( h , k ) = _________ ( h , k ) = _________

Equation: Equation:

Graph Linear Inequalities in Two Variables

After completing the Desmos Polygraph, think about what characteristics of the graphs from

the activity are similar or different from graphs you’ve seen before.

Were the graphs from the activity functions? Why or why not? If they were functions, state

the domain/range.

Linear inequality in two variables: replace the ‘=’ sign in a linear equation with <. , >, or .

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Example 1: Determine whether the ordered pair is a solution of the inequality

a. 3x – 4y > 9 (2, 0)

b. 2x + 3y

14 (5, 2)

c. y

8 (-9, -7)

Steps for graphing a linear inequality in two variables:

Step 1: Put the inequality into a nice graphing formant, and then graph the boundary

line.

Use a ____________ line for < or >

Use a ____________ line for ≤ or ≥

Step 2: Determine which side of the line to shade, and then shade that entire region.

Shade ____________ the line for < or ≤

Shade ____________ the line for > or ≥

Example 2: Graph the inequality

a. y < -

1

2x + 4 b. y

3x + 1

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c. 2y + 4x > 8 d. x + 4y < - 8

e. y < 2 f. x

1

e. y > -2x + 3 f. x

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