1. CCSS Content Standards F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Mathematical Practices 1 Make sense of problems and persevere in solving them.

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

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

4. 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).

5. 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.

6. Example 1 Piecewise-Defined Function Answer:

7. 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}.

8. 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

9. 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

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

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

12. 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.

13. 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.

14. Example 3 Use a Step Function Answer:

15. 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.

16. Concept

17. Example 4 Absolute Value Functions Graph y = |x| + 1. Identify the domain and range. Create a table of values. x|x| + 1 –34 –23 –12 01 12 23 34

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

19. 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.

20. 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.