LESSON 2–6 Special Functions

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

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

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

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

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

  TEKS

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

Concept

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

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

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

LESSON 2–6 Special Functions