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Graph Transformations 1

The graph of modified functions can often be deduced from simple geometrical transformations of the graph of the original function. To deduce the transformation, we often consider how a few points on the graph are modified from the original graph and then extrapolate.

We explain which modified function corresponds to the appropriate transformation on the graph of $$f$$. $$a\ne 0$$ is a fixed number.

  • Horizontal translation of $$a$$: $$f(x-a)$$
  • Vertical translation of $$a$$: $$f(x)+a$$
  • Horizontal scaling of ratio $$1/a$$: $$f(ax)$$. It is a stretch or dilation if $$\vert a\vert \lt 1$$ and a compression if $$\vert a\vert \gt 1$$.
  • Vertical scaling of ratio $$1/a$$: $$af(x)$$
  • Horizontal reflection (along vertical axis): $$f(-x)$$
  • Vertical reflection (along horizontal axis): $$-f(x)$$
  • Rotation of angle $$\pi$$ about the origin:$$-f(-x)$$
Transformation of graph of $$f(x)$$ Translation $$f(x-1)$$ horizontal scaling $$f(2x)$$ vertical scaling $$2f(x)$$ horizontal reflection $$f(-x)$$ vertical reflection $$-f(x)$$ symmetry $$-f(-x)$$
Transformation of graph of $$f(x)$$ Translation $$f(x-1)$$ horizontal scaling $$f(2x)$$ vertical scaling $$2f(x)$$ horizontal reflection $$f(-x)$$ vertical reflection $$-f(x)$$ symmetry $$-f(-x)$$