SpecialFunctionsLesson

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Slide 1
 Special Functions:
 Step, Absolute Value, and Piecewise Functions

Slide 2
 Ponder This…
 Suppose you took the advice of these guys and bought a ton of Duct Tape to fix everything that ails for life. If they come in 100foot rolls, can we find a function that takes the length of duct tape needed as our input and outputs how many rolls to buy? What would such a graph look like when graphed?

Slide 3
 Ponder This…
 This is what our ducttape function looks like. (What do we need to be careful about here to make sure it is a function?)
 We have never seen anything like this before! What witchery is this? Today, we will find out!

Slide 4  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.7b Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.
 Mathematical Practices
 4 Model with mathematics.
 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

Slide 5  Concept

Slide 6  Concept
 This is a graph of the greatest integer function. In this graph, y is equal to the greatest integer value of x. So, if x = 5.5, y = 5. Drag the slider to change the value of the green point on this function. 1. What happens when the green point approaches the open points? The closed points?2. Can x and y both be integers?3. Is y ever greater than x?4. Could you trace this graph without picking your pencil up off a piece of paper?5. How would you descibe this graph to someone else?
 T. I.P.S.!!!
 TIME:
 7 MIN

Slide 7  Example 1
 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:

Slide 8  Example 1
 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:

Slide 9  Example 1
 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.

Slide 10  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

Slide 11  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

Slide 12  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

Slide 13  Concept
 Check It Out!

Slide 14  Concept

Slide 15  Example 3
 A. D = all real numbers, R = all numbers ≥ 0
 B. D = all numbers ≥ 0R = all real numbers,
 D = all numbers ≥ 0, R = all numbers ≥ 0
 D = all real numbers, R = all real numbers
 Graph f(x) = │x + 3│. State the domain and range.

Slide 16  Example 3
 A. D = all real numbers, R = all numbers ≥ 0
 B. D = all numbers ≥ 0R = all real numbers,
 D = all numbers ≥ 0, R = all numbers ≥ 0
 D = all real numbers, R = all real numbers
 Graph f(x) = │x + 3│. State the domain and range.

Slide 17  Example 3
 A. D = all real numbers, R = all numbers ≥ 0
 B. D = all numbers ≥ 0R = all real numbers,
 D = all numbers ≥ 0, R = all numbers ≥ 0
 D = all real numbers, R = all real numbers
 Graph f(x) = │x + 3│. State the domain and range.

Slide 18  Concept
 Check It Out!

Slide 19  Example 4
 PiecewiseDefined Function

Slide 20  Example 4
 PiecewiseDefined 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).

Slide 21  Example 4
 PiecewiseDefined Function
 Answer:

Slide 22  Example 4
 PiecewiseDefined Function
 Answer:
 D = all real numbers, R = all real numbers

Slide 23  Concept
 T. I.P.S.!!!
 TIME:
 5 MIN

Slide 24  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

Slide 25  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

Slide 26  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

Slide 27  Concept
 Check It Out!

Slide 28  Concept