Special Functions: Step, Absolute Value, and Piecewise Functions.

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Presentation transcript:

Special Functions: Step, Absolute Value, and Piecewise Functions

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 100-foot 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?

Ponder This… This is what our duct-tape 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!

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 piecewise- defined functions, including step functions and absolute value functions. Mathematical Practices 4 Model with mathematics. Common Core State Standards © Copyright National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

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

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:

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:

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.

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 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 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

Concept Check It Out!

Concept

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

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

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

Concept Check It Out!

Example 4 Piecewise-Defined Function

Example 4 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 Piecewise-Defined Function Answer:

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

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

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

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

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

Concept Check It Out!

Concept