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Do Now

Do Now. Determine algebraically whether the function is even, odd, or neither. f(x) = x 6 – 2x 2 + 3 h(x) = x 3 – 5 f(x) = x √x + 5. 1.3 Shifting, Reflecting, and Stretching Graphs. Section Objectives : Recognize graphs of common functions.

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Do Now

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  1. Do Now • Determine algebraically whether the function is even, odd, or neither. • f(x) = x6 – 2x2 + 3 • h(x) = x3 – 5 • f(x) = x √x + 5

  2. 1.3 Shifting, Reflecting, and Stretching Graphs Section Objectives: Recognize graphs of common functions. Use vertical and horizontal shifts and reflections to sketch graphs of functions. Use nonrigid transformations to sketch graphs of functions.

  3. Today’s Objectives: • Identify vertical and horizontal shifts from an equation. • Write an equation for a function that has been translated vertically and/or horizontally.

  4. Why is this important? • Understanding shifts and reflections will help us quickly sketch graphs by hand.

  5. Before we start… • Pair up with one or two people sitting next to you. You shouldn’t have to move out of your seat to work with your partners.

  6. With your partner, sketch each graph in your notebook. • Match your graph up with the type of function. • Everyone needs to have the sketches in their notebooks. • f(x) = c (c is any number) Quadratic • f(x) = x Cubic • f(x) = |x| Constant • f(x) = √x Identity • f(x) = x2 Absolute Value • f(x) = x3 Square Root

  7. Vocabulary A special relation in which there is only one output for every input. Consists of a basic function and all of its transformations. All functions in a family have similar features. The most basic form a function takes. The result of a transformation to a parent function. Any alteration of the parent function. Moving a graph side to side or up and down. Does not alter the shape of a graph.

  8. Using your calculator… • Graph f(x) = x2 • Graph h(x) = x2 + 2 • What happened to the graph?

  9. Using your calculator… • Graph f(x) = x2 • Graph h(x) = x2 – 2 • What happened to the graph?

  10. Vertical Translations • Shifts of the graph either up or down • Adding a constant to the parent function shifts the graph up, y = f(x) + c • Subtracting a constant to the parent function shifts the graph down, y = f(x) - c

  11. Example • Identify the translation and then graph the function by translating the parent function. y = |x| - 2

  12. Example 2. Identify the translation and then graph the function by translating the parent function. y =

  13. Example 3. Write an equation for the translation of the identity function down 4 units.

  14. Practice Identify the translation and then graph the function by translating the parent function. • y = |x| + 1 • y = x2 – 4 • y = + 2 • y = x3 – 5 • y = + 3

  15. Practice • Write an equation for the translation of y =|x| up 1 unit • Write an equation for the translation of y = x2, up 3 units • Write an equation for the translation of y = down 5 units

  16. Using your calculator… • Graph f(x) = x2 • Graph h(x) = (x – 2)2 • What happened to the graph?

  17. Using your calculator… • Graph f(x) = x2 • Graph h(x) = (x + 2)2 • What happened to the graph?

  18. Horizontal Translations • Shifts of the graph either left or right • Adding a constant to the input of the parent function shifts the graph left, y = f(x + c) • Subtracting a constant to the parent function shifts the graph right, y = f(x – c)

  19. Example • Identify the translation and then graph the function by translating the parent function. y = |x – 2|

  20. Example • Identify the translation and then graph the function by translating the parent function. y = (x + 3)3

  21. Example 3. Write an equation for the translation of the square root of x left 4 units.

  22. Practice Identify the translation and then graph the function by translating the parent function. • y = • y = (x – 4)3 • y = |x – 5| • y = (x + 5) 2

  23. Practice • Write an equation for the translation of y = x2 right one unit 1 unit • Write an equation for the translation of y=|x| left 3 units • Write an equation for the translation of y = right 5 units

  24. Putting it all Together • Function might have both horizontal and vertical shifts • Remember to think about the inside and outside transformations • Inside is always horizontal, outside is always vertical

  25. Example 1. Graph the function y = |x – 3| + 1

  26. Example 2. Write an equation for the translation of y = x3 2 units down and 3 units left

  27. Practice • Graph the function y = (x + 5)2 + 3 • Graph the function y = • Graph the function y = |x + 2| - 4 • Write an equation for the translation of y = x2 1 unit down and 7 units right 5. Write an equation for the translation of y = x3 4 units up and 3 units right

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