1. Piecewise Functions 6-3 Warm Up Lesson Presentation Lesson Quiz
Holt McDougal Algebra 2
Holt Algebra 2

2. Write the equation of each line in slope-intercept form.
Warm Up
Write the equation of each line in slope-intercept form.
1. slope of 3 and passes through the point (5, 30)
y = 3x + 2 1 2
2. slope of – and passes through the point (6, 40)
y = – x + 43 1 2

3. Warm Up
For each function, determine whether the graph opens upward or downward.
1. f(x) = -4x2 + 6x + 1
downward
2. f(x) = 8x2 – x - 2
upward
Write each function in slope-intercept form.
3. Y + 3x =10
y = -3x + 10
4. -6y – 12x = 24
y = -2x - 4

4. Objectives Write and graph piecewise functions.
Use piecewise functions to describe real-world situations.

5. Vocabulary
piecewise function
step function

6. Essential Question
How can you tell where to put open and closed points on the graph of a piecewise function?

7. 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).

8. When using interval notation, square brackets [ ] indicate an included endpoint, and parentheses ( ) indicate an excluded endpoint. (Lesson A-1)
Remember!

9. Example 1: Consumer Application
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.

10. Example 1 Continued
The domain of the function is divided into three intervals:
Weights under 2
[0, 2)
Weights 2 and under 5
[2, 5)
[5, +∞)
Weights 5 and over

11. Example 1 Continued
Step 2 Write a verbal description.
Mixed nuts cost $8.00 per pound for less than 2 lb, $6.00 per pound for 2 lb or more and less than 5 lb, and $5.00 per pound for 5 or more pounds.

12. Check It Out! Example 1
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.

13. Check It Out! Example 1 Continued
The domain of the function is divided into three intervals:
Green Fee ($)
Time Range (h) 28
8 A.M. – noon 24
noon – 4 P.M. 12
4 P.M. – 9 P.M.
$28
[8, 12)
$24
[12, 4)
[4, 9)
$12

14. Check It Out! Example 1 Continued
Step 2 Write a verbal description.
The green fee is $28 from 8 A.M. up to noon, $24 from noon up to 4 P.M., and $12 from 4 up to 9 P.M.

15. 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. The rule for the movie ticket prices from Example 1 on page 422 is shown.

16. 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.”

17. 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.

18. Example 2A: Evaluating a Piecewise Function
Evaluate each piecewise function for x = –1 and x = 4.
2x if x ≤ 2
h(x) =
x2 – 4 if x > 2
Because –1 ≤ 2, use the rule for x ≤ 2.
h(–1) = 2(–1) + 1 = –1
Because 4 > 2, use the rule for x > 2.
h(4) = 42 – 4 = 12

19. Example 2B: Evaluating a Piecewise Function
Evaluate for x = -1 and x = 4
2x if x ≤ –1
g(x) =
5x if x > –1
g(–1) = 2(–1) = 1 2
Because –1 ≤ –1, use the rule for x ≤ –1.
Because 4 > –1, use the rule for x > –1.
g(4) = 5(4) = 20

20. Check It Out! Example 2a
Evaluate each piecewise function for x = –1 and x = 3.
12 if x < –3
f(x) =
15 if –3 ≤ x < 6
20 if x ≥ 6
Because –3 ≤ –1 < 6, use the rule for –3 ≤ x < 6 .
f(–1) = 15
Because –3 ≤ 3 < 6, use the rule for –3 ≤ x < 6 .
f(3) = 15

21. Check It Out! Example 2b
Evaluate each piecewise function for x = –1 and x = 3.
3x if x < 0
g(x) =
5x – if x ≥ 0
Because –1 < 0, use the rule for x < 0.
g(–1) = 3(–1)2 + 1 = 4
Because 3 ≥ 0, use the rule for x ≥ 0.
g(3) = 5(3) – 2 = 13

22. Essential Question
How can you tell where to put open and closed points on the graph of a piecewise function?
An open endpoint indicates a value not included in the domain and a closed endpoint indicates a value that is included in the domain.

23. Teacher: This piecewise function begins with high profit but shows big losses as it continues.
Student: Sound like a piece-unwise function to me.