1. LESSON 2–6
Special Functions

2. Five-Minute Check (over Lesson 2–5) TEKS Then/Now New Vocabulary
Example 1: Piecewise-Defined Function
Example 2: Write a Piecewise-Defined Function
Example 3: Real-World Example: Use a Step Function
Key Concept: Parent Functions of Absolute Value Functions
Example 4: Absolute Value Functions
Lesson Menu

3. Which scatter plot represents the data shown in the table?
C. D.
5-Minute Check 1

4. Which prediction equation represents the data shown in the table?
A. y = 2x + 94
B. y = 2x + 64
C. y = –2x + 94
D. y = –2x + 64
5-Minute Check 2

5. Use your prediction equation to predict the missing value.
B. $72
C. $82
D. $92
5-Minute Check 3

6. The scatter plot shows the number of summer workouts the players on a basketball team attended and the number of wins during the following season. Predict the number of wins the team would have if they attended 24 summer workouts.
A. 6
B. 12
C. 24
D. 48
5-Minute Check 4

7. TEKS

8. You modeled data using lines of regression.
Write and graph piecewise-defined functions.
Write and graph step and absolute value functions.
Then/Now

9. piecewise-defined function
piecewise-linear function
step function
greatest integer function
absolute value function
Vocabulary

10. Piecewise-Defined Function
Step 1 Graph the linear function f(x) = x – 1 for x ≤ 3. Since 3 satisfies this inequality, begin with a closed circle at (3, 2).
Example 1

11. Piecewise-Defined Function
Step 2 Graph the constant function f(x) = –1 for x > 3. Since x does not satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right.
Example 1

12. Piecewise-Defined Function
Answer: The function is defined for all values of x, so the domain is all real numbers. The values that are y-coordinates of points on the graph are all real numbers less than or equal to 2, so the range is {f(x) | f(x) ≤ 2}.
Example 1

13. A. domain: all real numbers range: all real numbers
B. domain: all real numbers range: {y|y > –1}
C. domain: all real numbers range: {y|y > –1 or y = –3}
D. domain: {x|x > –1 or x = –3} range: all real numbers
Example 1

14. Write the piecewise-defined function shown in the graph.
Write a Piecewise-Defined Function
Write the piecewise-defined function shown in the graph.
Examine and write a function for each portion of the graph.
The left portion of the graph is a graph of f(x) = x – 4. There is a circle at (2, –2), so the linear function is defined for {x | x < 2}.
The right portion of the graph is the constant function f(x) = 1. There is a dot at (2, 1), so the constant function is defined for {x | x ≥ 2}.
Example 2

15. Write the piecewise-defined function. Answer:
Write a Piecewise-Defined Function
Write the piecewise-defined function.
Answer:
Example 2

16. Identify the piecewise-defined function shown in the graph.B. C. D.
Example 2

17. Use a Step Function
PSYCHOLOGY One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.
Analyze The total charge must be a multiple of $85, so the graph will be the graph of a step function.
Formulate If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on.
Example 3

18. Use a Step Function
Determine Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph.
Example 3

19. Use a Step Function
Answer:
Justify Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint.
Example 3

20. same way. Our graph accurately represents the described situation.
Use a Step Function
Evaluate Though a step function can be represented by a single expression, the table allows us to break the expression into pieces based on the time intervals.
The table resembles a piecewise-defined function and can be graphed in the
same way. Our graph accurately represents the described situation.

21. SALES The Daily Grind charges $1
SALES The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation.
A. B.
C. D.
Example 3

22. Concept

23. Graph y = |x| + 1. Identify the domain and range.
Absolute Value Functions
Graph y = |x| + 1. Identify the domain and range.
Create a table of values. x
|x| + 1 –3 4 –2 3 –1 2 1
Example 4

24. Graph the points and connect them.
Absolute Value Functions
Graph the points and connect them.
Answer:
The domain is all real numbers. The range is {y | y ≥ 1}.
Example 4

25. Identify the function shown by the graph.
A. y = |x| – 1
B. y = |x – 1| – 1
C. y = |x – 1|
D. y = |x + 1| – 1
Example 4

26. LESSON 2–6
Special Functions