SpecialFunctionsLesson

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SpecialFunctionsLesson

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Slide Content
  1. Slide 1

    • Special Functions:
    • Step, Absolute Value, and Piecewise Functions
  2. 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 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?
  3. Slide 3

    • 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!
  4. CCSS

    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 piecewise-defined 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.
  5. Concept

    Slide 5 - Concept

  6. 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
  7. Example 1

    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:
  8. Example 1

    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:
  9. Example 1

    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.
  10. Example 1

    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
  11. Example 1

    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
  12. Example 1

    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
  13. Concept

    Slide 13 - Concept

    • Check It Out!
  14. Concept

    Slide 14 - Concept

  15. Example 3

    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.
  16. Example 3

    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.
  17. Example 3

    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.
  18. Concept

    Slide 18 - Concept

    • Check It Out!
  19. Example 4

    Slide 19 - Example 4

    • Piecewise-Defined Function
  20. Example 4

    Slide 20 - 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).
  21. Example 4

    Slide 21 - Example 4

    • Piecewise-Defined Function
    • Answer:
  22. Example 4

    Slide 22 - Example 4

    • Piecewise-Defined Function
    • Answer:
    • D = all real numbers, R = all real numbers
  23. Concept

    Slide 23 - Concept

    • T. I.P.S.!!!
    • TIME:
    • 5 MIN
  24. Example 4

    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
  25. Example 4

    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
  26. Example 4

    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
  27. Concept

    Slide 27 - Concept

    • Check It Out!
  28. Concept

    Slide 28 - Concept