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Function Transformations

Learn how shifts, reflections, stretches, and compressions alter graphs with specific parameter changes to functions. Understand vertical and horizontal shifts, reflections, and vertical/horizontal stretches or compressions. Compare different transformations on f(x) including vertical shifts with +3, -2, horizontal shifts with ±3, ±1, x-axis reflections, and vertical/horizontal stretches or compressions by factors like 2 and 4. Get insights into how parameter changes influence the graph positioning, shape, and orientation.

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Function Transformations

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  1. Function Transformations Goal(s): Analyze the effects on a graph when the parameters of the equation are changed Vertical shifts Horizontal shifts Reflections Vertical stretches or compressions Horizontal stretches or compressions

  2. Vertical Shifts Given a graph f(x): Compare it to f(x) + 3:

  3. Vertical Shifts Given a graph f(x): Compare it to f(x) – 2:

  4. Vertical Shifts In general, a graph f(x) + k is the graph of f(x) shifted up(+) or down(-) k units. So f(x) + 5is shifted up 5 units from f(x) and f(x) – 8is shifted down 8 units from f(x).

  5. Horizontal Shifts Given a graph f(x): Compare it to f(x + 3):

  6. Horizontal Shifts Given a graph f(x): Compare it to f(x – 1):

  7. Horizontal Shifts In general, a graph f(x - h) is the graph of f(x) shifted left(+) or right(-) h units. So f(x + 5)is shifted left 5 units from f(x) and f(x – 8)is shifted right 8 units from f(x).

  8. Reflection over the x-axis Given a graph f(x): Compare it to -f(x): In general, a graph -f(x) is the graph of f(x) reflected over the x-axis.

  9. Vertical Stretch or Compression Given a graph f(x): Compare it to 2f(x):

  10. Vertical Stretch or Compression Given a graph f(x): Compare it to f(x):

  11. Vertical Stretch or Compression In general, a graph af(x) is the graph of f(x) vertically stretched or compressed. If a<1, there is a compression If a >1, there is a stretch So f(x)is vertically compressed by a factor of ½ and 4f(x)is vertically stretched by a factor of 4

  12. Horizontal Stretch or Compression Given a graph f(x): Compare it to f(3x):

  13. Horizontal Stretch or Compression Given a graph f(x): Compare it to f(x):

  14. Horizontal Stretch or Compression In general, a graph f(bx) is the graph of f(x) horizontally stretched or compressed. If b<1, there is a stretch If b >1, there is a compression So f(x)is horizontally stretched by a factor of 2 and f(4x)is horizontally compressed by a factor of

  15. Function Transformations • Given a function, f(x) the following are general transformations of the graph of the function: -af(b(x-h))+k • h  horizontal shift (left or right) • b  horizontal stretch or compression • a  vertical stretch or compression • (in front of function)  reflection over x-axis • k  vertical shift (up or down)

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