Piecewise FunctionsPre-Cal Advanced

Ms. Ouseph

Objective for today!

We will use our knowledge of domain

and transformations to graph piecewise

functions.

A piecewise function is…

A function defined by different equations on

different domains.

In other words…Think of it as only

having a “chunk” of a function!

We’ll harp more about that later…

For now we will explore an everyday situation where piecewise functions

are applicable.

arking Garage

Suppose we were parking in a parking garage that we had to pay for by the hour. The pay structure is as follows:

• For the first 4 hours, it costs us $3 per hour to park.• For any number of hours parked after 4 hours, it costs $2 per

hour to park(These rates are proportional to any amount of time spent parked within the first

three hours; for example if we only stayed for ½ hour, we would pay $1.50.)

OUR GOAL: Create a function which inputs hours parked and outputs total cost.

Total Hours Parked (x) Total Payment (y or f(x))

*For the first 4 hours, it costs us $3/hour to park*

12 34

Equation 1: *For any number of hours parked after 4 hours, it costs us $2/hour to park*

56 78x

Equation 2:

y = 3x

y = 2x + 412 + 2(x – 4)

f(1) = 3(1) = $3f(2) = 3(2) = $6f(3) = 3(3) = $9f(4) = 3(4) = $12

f(5) = 3(4) + 2(5-4) = $14f(6) = 3(4) + 2(6-4) = $16f(7) = 3(4) + 2(7-4) = $18f(8) = 3(4) + 2(8-4) = $20

In groups, fill out the table below and determine an equation to calculate the total payment with respect to the hourly rate.

Equation 1: y = 3x only works when…

Equation 2: y = 2x + 4 only works when…

Combining these two, we have a piecewise function!

The total number of hours parked (x) are between 0-4

0 ≤ x ≤ 4

The total number of hours parked (x) are between greater than 4

x > 4

f(x) = {3x, 0 ≤ x ≤ 4 2x + 4, x > 4

How do we graph a piecewise function??Using the table we created, we can plot points on a graph to graph

the piecewise function!

x y x y

01234

45678

0369

12

1214161820

y = 3x, 0 ≤ x ≤ 4 y = 2x + 4, x > 4

Closed

Open

Each function must live in its own “neighborhood”Let's put up a fence to separate these two “neighborhoods.”

y = 3xlives here!

y = 2x + 4lives here!

x = 4

Note: Its OK for each

neighborhood to lie on the fence. They just

can’t cross over it!

Total Hours Parked

Tota

l Pay

men

tParking Hours vs. Payment

(0, 0)(1, 3)(2, 6)(3, 9)(4, 12)

(4, 12)(5, 14)(6, 16)(7, 18)(8, 20)

Both equations are “on the fence,” but one point (from y=3x) is closed while

the other (from y=2x+4) is open!

f(x) = {3x, 0 ≤ x ≤ 4 2x + 4, x > 4

This analysis is not meant to explain the parking situation mathematically. Rather, it is meant to

explain the idea of a piecewise function by showing you where one exists in real life.

Now, we will be exploring and graphing other piecewise functions involving an array of

functions (linear, quadratic, etc.)!

What do we need to know in order to graph these pieces?

•

“Fence,” Domain, Points, Open/Closed, etc…

•

x y x y

10-1-2

1234

531-1

3210

Closed Open

Now let’s use these table of values to graph our piecewise function!

Domain: (-∞, ∞) Range: (-∞, 5]

x y x y10-1-2

531-1

1234

3210

Let’s look at a quadratic piecewise function!

Work with a partner to graph the following function: •

•

x y x y

0-1-2-3

0123

32-1-6

1234

( ) Closed Open ( )

Now let’s use these table of values to graph our piecewise function!

Domain: (-∞, ∞) Range: (-∞, ∞)

x y x y0123

1234

0-1-2-3

32-1-6

In summary…

• The key to graphing piecewise functions is:– Determining the “fence”

• This fence will tell you where each piece of the function “lives”

– Domain• Once you know the domain of each function, its

just a matter of creating a table of values to plot points!

Now that we have explored various piecewise functions,

you will be given time to finish the rest of your worksheet.