Transformations of power functions
Theory
A power function \(y=x^{n}\) — including the reciprocal \(\dfrac{1}{x}\), the reciprocal square \(\dfrac{1}{x^{2}}\) and the square root \(\sqrt{x}\) — can be dilated, reflected and translated into the form \(y=a(x-h)^{n}+k\). Here \(a\) is a dilation from the \(x\)-axis (with a reflection if \(a<0\)), \(h\) is a horizontal translation and \(k\) a vertical one, and from these you can sketch the curve and state its domain, range and asymptotes.
A power function has the form \(y=x^{n}\). As well as the polynomials \(x^{2},x^{3},x^{4},\dots\), this family includes the reciprocal \(\dfrac{1}{x}=x^{-1}\), the reciprocal square \(\dfrac{1}{x^{2}}=x^{-2}\), and the square root \(\sqrt{x}=x^{1/2}\). Each has a characteristic shape and a natural “base point” — the origin for \(x^{n}\), a pair of asymptotes for the reciprocals, and an endpoint for \(\sqrt{x}\).
Every such graph can be transformed by a combination of a dilation, a reflection and translations, giving the general rule \(y=a(x-h)^{n}+k\). The constant \(a\) stretches the graph by factor \(|a|\) from the \(x\)-axis (and reflects it in the \(x\)-axis when \(a<0\)); \(h\) slides it \(h\) units horizontally; and \(k\) slides it \(k\) units vertically. The reciprocal and root analogues are \(y=\dfrac{a}{x-h}+k\), \(y=\dfrac{a}{(x-h)^{2}}+k\) and \(y=a\sqrt{x-h}+k\).
The key features move with the graph. A turning or stationary point at the origin moves to \((h,k)\); the endpoint of a square-root curve moves to \((h,k)\) so its domain becomes \(x\ge h\); and the asymptotes of a reciprocal move to \(x=h\) and \(y=k\). Reading \(a,h,k\) off the rule lets you sketch the image and state its domain and range.
The general transformed power function — \(a\) dilates from the \(x\)-axis (reflection if \(a<0\)), \(h\) and \(k\) translate:
The reciprocal and reciprocal-square analogues, with vertical asymptote \(x=h\) and horizontal asymptote \(y=k\):
The square-root analogue, with endpoint \((h,k)\) and domain \(x\ge h\):
How to sketch a transformed power function
- Write it in standard form. Express the rule as \(y=a(x-h)^{n}+k\) (or \(\dfrac{a}{x-h}+k\), \(\dfrac{a}{(x-h)^{2}}+k\), \(a\sqrt{x-h}+k\)) and read off \(a\), \(h\) and \(k\).
- Dilate and reflect first. Stretch the base graph by factor \(|a|\) from the \(x\)-axis; if \(a<0\), reflect it in the \(x\)-axis.
- Then translate. Move the graph \(h\) units horizontally (right if \(h>0\)) and \(k\) units vertically (up if \(k>0\)). Shift the key feature — the turning/stationary point, the endpoint, or the asymptotes — by \((h,k)\).
- Find intercepts. Set \(x=0\) for the \(y\)-intercept and \(y=0\) for the \(x\)-intercept(s), where they exist.
- State domain and range. Use the shifted asymptotes (reciprocals) or the endpoint and direction (square root) to write the domain and range.
| \(h,\ k\) | \(=\) | \(2,\ 1\) |
| \((0,0)\) | \(\to\) | \((2,1)\) |
| \((x-2)^{3}+1\) | \(=\) | \(0\) |
| \((x-2)^{3}\) | \(=\) | \(-1\) |
| \(x-2\) | \(=\) | \(-1\) |
| \(x\) | \(=\) | \(1\) |
| \(y\) | \(=\) | \((-2)^{3}+1=-7\) |
| \(a,\ h,\ k\) | \(=\) | \(2,\ -1,\ -3\) |
| dilation | factor \(2\) from the \(x\)-axis | |
| translation | \(1\) left, \(3\) down |
| vertical | \(:\) | \(x=-1\) |
| horizontal | \(:\) | \(y=-3\) |
| \(a,\ h,\ k\) | \(=\) | \(-1,\ -3,\ 2\) |
| \((0,0)\) | \(\to\) | \((-3,2)\) |
| domain | \(:\) | \(x\ge -3\) |
| range | \(:\) | \(y\le 2\) |
| \(\sqrt{x+3}\) | \(=\) | \(2\) |
| \(x+3\) | \(=\) | \(4\) |
| \(x\) | \(=\) | \(1\) |
| \(y\) | \(=\) | \(-2(x+4)^{3}+5\) |
| \(a,\ h,\ k\) | \(=\) | \(-2,\ -4,\ 5\) |
| \(|a|=2\) | \(\Rightarrow\) | dilation factor \(2\) from the \(x\)-axis |
| \(a<0\) | \(\Rightarrow\) | reflection in the \(x\)-axis |
| \(h=-4\) | \(\Rightarrow\) | translate \(4\) left |
| \(k=5\) | \(\Rightarrow\) | translate \(5\) up |
Common pitfalls
Frequently asked questions
What do \(a\), \(h\) and \(k\) do in \(y=a(x-h)^{n}+k\)?
\(a\) is a dilation by factor \(|a|\) from the \(x\)-axis, and reflects in the \(x\)-axis if \(a<0\); \(h\) is a horizontal translation (right if \(h>0\)); and \(k\) is a vertical translation (up if \(k>0\)). Together they map \(y=x^{n}\) onto the transformed curve.
Which way does \(y=(x-h)^{n}\) move compared with \(y=x^{n}\)?
It moves horizontally by \(h\), opposite to the sign inside the bracket. So \(y=(x-2)^{3}\) is \(y=x^{3}\) shifted \(2\) right, while \(y=(x+2)^{3}\) is shifted \(2\) left.
How do you find the asymptotes of a transformed reciprocal function?
For \(y=\dfrac{a}{x-h}+k\), the vertical asymptote is \(x=h\) (denominator zero) and the horizontal asymptote is \(y=k\). The same holds for the reciprocal square \(y=\dfrac{a}{(x-h)^{2}}+k\).
What are the domain and range of \(y=a\sqrt{x-h}+k\)?
The root needs \(x-h\ge 0\), so the domain is \(x\ge h\) and the curve starts at \((h,k)\). If \(a>0\) the range is \(y\ge k\); if \(a<0\) it is reflected, so \(y\le k\).
In what order do you apply the transformations?
Dilations and reflections first, then translations. For \(y=a(x-h)^{n}+k\), scale by \(|a|\) and reflect if \(a<0\), then translate by \(h\) and \(k\). Translating first can place the graph wrongly.
How do you sketch a transformed power function?
Read \(a\), \(h\), \(k\) off the rule, move the key feature (turning point, endpoint or asymptotes) by \((h,k)\), apply the dilation and any reflection, find the intercepts, and state the domain and range.