Special Functions Lesson 9-7 Splash Screen LEARNING GOAL Understand how to identify and graph step functions, absolute value functions and piecewise-defined.

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Presentation transcript:

Special Functions Lesson 9-7 Splash Screen

LEARNING GOAL Understand how to identify and graph step functions, absolute value functions and piecewise-defined functions Then/Now

Vocabulary Vocabulary

Concept

Greatest Integer Function First, make a table of values. Select a few values between integers. On the graph, dots represent points that are included. Circles represent points that are not included. Answer: Because the dots and circles overlap, the domain is all real numbers. The range is all integers. Example 1

A. D = all real numbers, R = all real numbers B. D = all integers, R = all integers C. D = all real numbers, R = all integers D. D = all integers, R = all real numbers Example 1

Step Function TAXI A taxi company charges a fee for waiting at a rate of $0.75 per minute or any fraction thereof. Draw a graph that represents this situation. The total cost for the fee will be a multiple of $0.75, and the graph will be a step function. If the time is greater than 0 but less than or equal to 1 minute, the fee will be $0.75. If the time is greater than 2 minutes but less than or equal to 3 minutes, you will be charged for 3 minutes, or $2.25. Example 2

Step Function Answer: Example 2

SHOPPING An on-line catalog company charges for shipping based upon the weight of the item being shipped. The company charges $4.75 for each pound or any fraction thereof. Draw a graph of this situation. Example 2

A. B. C. Example 2

Concept

Graph f(x) = │2x + 2│. State the domain and range. Absolute Value Function Graph f(x) = │2x + 2│. State the domain and range. Since f(x) cannot be negative, the minimum point of the graph is where f(x) = 0. f(x) = │2x + 2│ Original function 0 = 2x + 2 Replace f(x) with 0. –2 = 2x Subtract 2 from each side. –1 = x Divide each side by 2. Example 3

Absolute Value Function Next, make a table of values. Include values for x > –5 and x < 3. Answer: The domain is all real numbers. The range is all nonnegative numbers. Example 3

Graph f(x) = │x + 3│. State the domain and range. A. D = all real numbers, R = all numbers ≥ 0 B. D = all numbers ≥ 0 R = all real numbers, D = all numbers ≥ 0, R = all numbers ≥ 0 D = all real numbers, R = all real numbers Example 3

Piecewise-Defined Function Graph the first expression. Create a table of values for when x < 0, f(x) = –x, and draw the graph. Since x is not equal to 0, place a circle at (0, 0). Next, graph the second expression. Create a table of values for when x ≥ 0, f(x) = –x + 2, and draw the graph. Since x is equal to 0, place a dot at (0, 2). Example 4

D = all real numbers, R = all real numbers Piecewise-Defined Function Answer: D = all real numbers, R = all real numbers Example 4

D = y│y ≤ –2, y > 2, R = all real numbers D = all real numbers, R = y│y ≤ –2, y > 2 D = all real numbers, R = y│y < –2, y ≥ 2 D. D = all real numbers, R = y│y ≤ 2, y > –2 Example 4

Example: Concept

Homework p. 602 #17-41 odd, Chapter 9 Review End of the Lesson