5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate

5.2 Piecewise FunctionsCC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Warm UpWrite the equation of each line in slope-intercept form.

1. slope of 3 and passes through the point (50, 200)

y = 3x + 50

2. slope of – and passes through the point (6, 40) 1 2

y = – x + 43 1 2


Write and graph piecewise functions.

Use piecewise functions to describe real-world situations.

Objectives


piecewise functionstep function

Vocabulary


A piecewise function is a function that is a combination of one or more functions. The rule for a piecewise function is different for different parts, or pieces, of the domain. For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).


When using interval notation, square brackets [ ] indicate an included endpoint, and parentheses ( ) indicate an excluded endpoint.

Introducing!


The domain of the function is divided into three intervals:

Weights under 2 [0, 2)

[2, 5)

[5, ∞)

Weights 2 and under 5

Weights 5 and over

Example 1


Create a table and a verbal description to represent the graph.

Example 1

Create a table

Because the endpoints of each segment of the graph identify the intervals of the domain, use the endpoints and points close to them as the domain values in the table.


Create a table and a verbal description to represent the graph.

Step 1 Create a table

Because the endpoints of each segment of the graph identify the intervals of the domain, use the endpoints and points close to them as the domain values in the table.

Example 2


The domain of the function is divided into three intervals:

$28 [8, 12)

[12, 4)

[4, 9)

$24

$12

Example 2 Continued

Green Fee ($)

Time Range (h)

28 8 A.M. – noon

24 noon – 4 P.M.

12 4 P.M. – 9 P.M.


A piecewise function that is constant for each interval of its domain, such as the ticket price function, is called a step function. You can describe piecewise functions with a function rule. Read this as “f of x is 5 if x is greater than 0 and less than 13, 9 if x is greater than or equal to 13 and less than 55, and 6.5 if x is greater than or equal to 55.”


2x + 1 if x ≤ 2

x2 – 4 if x > 2 h(x) =

Because –1 ≤ 2, use the rule for x ≤ 2.

Because 4 > 2, use the rule for x > 2.

h(–1) = 2(–1) + 1 = –1

h(4) = 42 – 4 = 12

Example 3

Evaluate each piecewise function for x = –1 and x = 4.

To evaluate any piecewise function for a specific input, find the interval of the domain that contains that input and then use the rule for that interval.


12 if x < –3

20 if x ≥ 6f(x) =

Because –3 ≤ –1 < 6, use the rule for –3 ≤ x < 6 .

f(–1) = 15

Example 4 Evaluate each piecewise function for x = –1 and x = 3.

15 if –3 ≤ x < 6

f(3) = 15 Because –3 ≤ 3 < 6, use the rule for –3 ≤ x < 6 .


Example 5

g(x) = 1 2

Graph each function.

x – 3 if 0 ≤ x < 4

x2 – 3 if x < 0

The function is composed of one linear piece and two quadratic pieces. The domain is divided at x = 0 and at x = 4.

(x – 4)2 – 1 if x ≥ 4


Example 5 Continued

No circle is required at (0, –3) and (4, –1) because the function is connected at those points.


O

●

The function is 4 when x ≤ –1, so plot the point (–1, 4) with a closed circle and draw a horizontal ray to the left. The function is –2 when x > –1, so plot the point (–1, –2) with an open circle and draw a horizontal ray to the right.

f(x) =

Graph the function.

4 if x ≤ –1

–2 if x > –1

Example 6


Example 6: Sports Application

Jennifer is completing a 15.5-mile triathlon. She swims 0.5 mile in 30 minutes, bicycles 12 miles in 1 hour, and runs 3 miles in 30 minutes. Sketch a graph of Jennifer’s distance versus time. Then write a piecewise function for the graph.


Step 1 Make a table to organize the data. Use the distance formula to find Jennifer’s rate for each leg of the race.

Example 6 Continued


Step 2 Because time is the independent variable, determine the intervals for the function.

Swimming: 0 ≤ t ≤ 0.5

Biking: 0.5 < t ≤ 1.5

Running: 1.5 < t ≤ 2

She swims for half an hour.

She bikes for the next hour.

She runs the final half hour.

Example 6 Continued


Step 3 Graph the function.

After 30 minutes, Jennifer has covered 0.5 miles. On the next leg, she reaches a distance of 12 miles after a total of 1.5 hours. Finally she completes the 15.5 miles after 2 hours.

Example 6 Continued


Step 4 Write a linear function for each leg.

Use point-slope form: y – y1 = m(x – x1).

Swimming: d = t

Biking: d = 12t – 5.5

Running: d = 6t + 3.5

Use m = 0.5 and (0, 0).

Use m = 12 and (0.5, 0.5).

Use m = 6 and (1.5, 12.5).

The function rule is d(t) =

t if 0 ≤ t ≤ 0.5

12t – 5.5 if 0.5 < t ≤ 1.5

6t + 3.5 if 1.5 < t ≤ 2

Example 6 Continued


Independent Practice

Due Tomorrow at Beginning of Class

p. 426 # 9-18 all


Lesson Quiz: Part I

1. Graph the function, and evaluate at x = 1 and x = 3.

p(x) =x2 + 2 if x ≤ 2

1 2

x + 3 if x > 2 1 2


Lesson Quiz: Part II2. Write and graph a piecewise function for the

following situation. A house painter charges $12 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that. How much does he earn for a 70-hour week?

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate