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Precalc 3.4 - Shifting, Reflecting and Stretching Graphs

f(x) = c. f(x) = x. f(x) = |x|. f(x) = x. f(x) = x 2. f(x) = x 3. Precalc 3.4 - Shifting, Reflecting and Stretching Graphs. Shifting, Reflecting and Stretching = “ Transformations ”. Basic Graphs:. Outside change: f(x) + 2 = x 2 + 2. Inside change: f(x + 2) = (x + 2) 2.

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Precalc 3.4 - Shifting, Reflecting and Stretching Graphs

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  1. f(x) = c f(x) = x f(x) = |x| f(x) = x f(x) = x2 f(x) = x3 Precalc 3.4 - Shifting, Reflecting and Stretching Graphs • Shifting, Reflecting and Stretching = “Transformations” • Basic Graphs:

  2. Outside change: f(x) + 2 = x2 + 2 Inside change: f(x + 2) = (x + 2)2 f(x)= (x + 2)2 f(x) = x2 f(x) = x2 + 2 Vertical and Horizontal Shifting • Changes inside the function vs. changes outside the function • Addition / Subtraction = a vertical or horizontal SHIFT ex: f(x) = x2

  3. Shifting + Inside Outside = x3 shifted one unit upward = x3 shifted one unit to the right = x3 shifted two units left and oneunit up (Shifting cont’d…) Quick Reference Table: ex: if f(x) = x3, describe the following transformations: g(x) = x3 + 1 h(x) = (x - 1)3 k(x) = (x + 2)3 + 1

  4. ex: f(x) = x Outside negative = -[f(x)] = - x Inside negative = f(-x) = -x Reflection • Mirroring across the x or y axis • Produced by negative signs • Outside negative = x-axis / inside negative = y-axis • (Remember: Outside = “X-ternal”)

  5. ex: f(x) = - x + 2 Outside Negative Inside Addition Mirroring cont’d = Graph will reflect across the x-axis and move two units left…

  6. x2 (for c0) ex: f(x) = c Stretching • “Non-rigid” transformation • Position vs. Distortion • Accomplished with division and multiplication • Only concerned with vertical distortion • Division = “Shrink” / Multiplication = “Stretch” = Vertical shrink “by a factor of c” ex: f(x) = cx2 = Vertical stretch “by a factor of c”

  7. x2 2x2 x2/2

  8. f(-2) f(2) 3.4-A: Even and Odd Symmetry • A function is even if, for each x in the domain of f, f(-x) = f(x) ex: f(x) = x2 = = 4 4 (-2, 4) (2, 4) -2 2

  9. f(-1) = -[f(1)] -1 1 • A function is odd if, for each x in the domain of f, f(-x) = -[f(x)] ex: f(x) = x3 1 = -1 -1

  10. y y y Even / Odd Lines of Symmetry Even functions are symmetric ‘with respect to’ the Y axis…

  11. y = x y y y Odd functions are symmetric with respect to the ORIGIN…

  12. g(-x) = (-x)3 - (-x) Opposite input produces opposite output Algebraic Testing for Symmetry ex: Determine whether the following functions are even, odd or neither… 1. g(x) = x3 - x = -x3 + x = -(x3 - x) = -[g(x)] Numeric Check: g(2) = 6 ---> g(-2) = -6 = ODD

  13. h(-x) = (-x)2 + 1 Opposite input produces same output = h(x) = x2 + 1 2. h(x) = x2 + 1 Numeric Check: g(2) = 5 ---> g(-2) = 5 = EVEN

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