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Warmups :

Warmups : . You may use your graphing calculators for the following: Compare/Contrast the graphs of y = x 2 , y = 3x 2 , and y = 5x 2 Compare/Contrast the graphs of y = ⅛x 2 and y = ¼x 2 Compare/Contrast the graphs of y = -x 2 , y = -4x 2 , and y = -⅛x 2.

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Warmups :

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  1. Warmups: • You may use your graphing calculators for the following: • Compare/Contrast the graphs of • y = x2, y = 3x2, and y = 5x2 • Compare/Contrast the graphs of • y = ⅛x2 and y = ¼x2 • Compare/Contrast the graphs of • y = -x2, y = -4x2, and y = -⅛x2 These graphs represent vertical stretch and reflection translations.

  2. Beyond Plotting Points ~ Vertical and Horizontal Stretch (6.1) To make it easier … let’s look at a function that we’re comfortable with … y = x2 x y So … (2, 4) and (-2, 4) are points on the graph y = x2… 2 4 -2 4

  3. Original points on y = x2: (2, 4) and (-2, 4) COMPARE: y = 2x2 and y = (2x)2 Multiply the input by 2. Multiply the output by 2. Vertical Stretch Horizontal Stretch ** look for the same inputs as f(x) ** look for the same outputs as f(x) Y = 2f(2) = 8 2 = 2x 1 = x 1= x ( 2, 8) ( 1, 4) New points: New points: (-2, 8) ( -1, 4) Same inputs as original Same outputs as original

  4. Example: x -3 -2 -1 0 1 2 3 f(x) 3 2 1 0 1 2 3 a) Find 2f(x) Check w/ graphing calculator: f(x) = |x| Y1 = |x| Y2 = 2|x| Y3 = |2x| Use the table to look at specific values -3 -2 -1 0 1 2 3 x 2f(x) 6 4 2 0 2 4 6 b) Find f(2x) - 1 -1/2 0 1/2 1 3/2 -3/2 x f(2x) 3 2 1 0 1 2 3

  5. Odd-Even Functions Even ~ A function is said to be evenif f(-x) always yields f(x) A function is said to be oddif f(-x) always yields -f(x) Odd ~

  6. Example: Is f(x) = x2 is an even function, an odd function, or neither? FYI: Any EVEN function is symmetric to the y-axis. On a smaller scale … IN: 2 á OUT: 4 IN: -2 OUTOO á OUT: 4 f(x) = x2 Results in the same output … Therefore, EVEN function. f(-x) = (-x)2 = x2

  7. Example: Is f(x) = x3 is an even function, an odd function, or neither? FYI: Any ODD function is symmetric to the origin On a smaller scale … IN: 2 á OUT: 8 IN: -2 OUTOO á OUT: -8 Meaning each pre-image point is the same distance to the origin as its image point f(x) = x3 Results in negative outputs of each other … Therefore, ODD function. f(-x) = (-x)3 = -x3

  8. Review: What does y = a(x – h)2 + k mean?

  9. GROUPS: • Each group makes a poster of a parent function: • The poster must include: • Equation of the parent function • 5 points in a data table • graph that includes the 5 points • Domain, Range, Zeros, f(0) =____ • Increasing/Decreasing intervals •  For what values of x, does f(x) increase? •  For what values of x, does f(x) decrease? • Even, Odd, Neither? • Posters will be graded on the above as well as neatness!

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