Notes
Before you start: read a graph's domain and range and rearrange a linear equation.
Changes outside \(f\) act on outputs: \(af(x)+b\) maps \(y\) to \(ay+b\). Changes inside act on inputs: for \(f(cx+d)\), a point with old input \(u\) has new input \((u-d)/c\), provided \(c\ne0\).
\[y=af(cx+d)+b:\qquad (u,v)\longmapsto\left(\frac{u-d}{c},av+b\right).\]Thus \(f(x-h)\) moves a graph right by \(h\); \(f(cx)\) gives a horizontal scale factor \(1/|c|\), with reflection in the y-axis if \(c<0\). A negative outside multiplier reflects in the x-axis. Describe an ordered sequence or give an unambiguous coordinate rule.
Find the new domain by putting \(cx+d\) in the old domain. Find the new range by applying \(ay+b\) to the old range; reverse the endpoint order if \(a<0\).
Recall later
Under \(y=2f(x-1)+3\), the point \((4,-2)\) moves to \((5,-1)\).
Common mistakes
- Using the inside scale factor instead of its reciprocal.
- Translating vertically before stretching when the given formula requires the other order.
- Forgetting to transform domain or range endpoints.