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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 $-f(-x)$