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Lesson Menu Five-Minute Check (over Lesson 2–5) 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

Over Lesson 2–5 A.A B.B C.C D.D 5-Minute Check 1 Which scatter plot represents the data shown in the table? A.B. C.D.

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

Over Lesson 2–5 A.A B.B C.C D.D 5-Minute Check 3 A.$62 B.$72 C.$82 D.$92 Use your prediction equation to predict the missing value.

Over Lesson 2–5 A.A B.B C.C D.D 5-Minute Check 4 A.6 B.12 C.24 D.48 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.

Then/Now You modeled data using lines of regression. (Lesson 2–5) Write and graph piecewise-defined functions. Write and graph step and absolute value functions.

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

Example 1 Piecewise-Defined Function Step 1Graph 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 2Graph 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 {y | y ≤ 2}.

A.A B.B C.C D.D 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 2 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 a Piecewise-Defined Function Write the piecewise-defined function. Answer:

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

Example 3 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. Understand The total charge must be a multiple of $85, so the graph will be the graph of a step function. Plan 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 SolveUse 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: Check 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.

A.A B.B C.C D.D Example 3 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.

Concept

Example 4 Absolute Value Functions Graph y = |x| + 1. Identify the domain and range. Create a table of values. x|x| + 1 –34 –23 –

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

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

End of the Lesson