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Learn how to graph, analyze, and write piecewise-defined and absolute value functions with real-world examples.

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  1. Five-Minute Check (over Lesson 2–4) 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

  2. You modeled data using lines of regression. • Graph and analyze piecewise-defined functions. • Graph and analyze step and absolute value functions. Then/Now

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

  4. 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

  5. Piecewise-Defined Function Step 2Graph the constantfunction f(x)= –1 forx > 3. Since x doesnot satisfy thisinequality, begin withan open circle at(3, –1) and draw ahorizontal ray to theright. Example 1

  6. Piecewise-Defined Function Answer: The function is defined for all values of x, so the domain is all realnumbers. The values that arey-coordinates of points on thegraph are all real numbersless than or equal to 2, so therange is {f(x)| f(x)≤ 2}. Example 1

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

  8. 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

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

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

  11. 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

  12. Use a Step Function Solve 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

  13. 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. Example 3

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

  15. Concept

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

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

  18. 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

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