Translations
Theory
A translation slides a graph horizontally and/or vertically without changing its shape or size. Writing \(y=f(x-h)\) shifts the graph \(h\) units horizontally (right when \(h>0\)), \(y=f(x)+k\) shifts it \(k\) units vertically (up when \(k>0\)), and the combined form \(y=f(x-h)+k\) maps every point \((x,y)\mapsto(x+h,\,y+k)\).
A translation is a transformation that slides a graph to a new position without changing its shape, size or orientation. The image is congruent to the original — every point moves by the same horizontal and vertical amount, so intercepts, turning points and asymptotes all move together.
A vertical translation is produced by adding a constant: \(y=f(x)+k\) moves the graph up by \(k\) when \(k>0\) and down when \(k<0\). A horizontal translation comes from replacing \(x\) with \(x-h\): \(y=f(x-h)\) moves the graph right by \(h\) when \(h>0\) and left when \(h<0\). The horizontal shift looks “backwards” because the bracket must equal the original input, so \(x\) has to be larger.
Combining both gives \(y=f(x-h)+k\), under which each point transforms by the mapping \((x,y)\mapsto(x+h,\,y+k)\). Applying this rule to the key points of a standard curve — and shifting any asymptotes, the domain and the range by the same amounts — is the fastest way to sketch the image.
The general translated graph — \(h\) horizontal, \(k\) vertical:
The point mapping — every point moves by \((h,k)\):
Asymptotes shift by the same amounts — a vertical asymptote by \(h\), a horizontal asymptote by \(k\):
How to translate a graph and write its equation
- Read off \(h\) and \(k\). A horizontal shift of \(h\) (right positive) replaces \(x\) with \(x-h\); a vertical shift of \(k\) (up positive) adds \(k\).
- Write the rule. Substitute into \(y=f(x-h)+k\), e.g. “\(y=x^2\), right \(2\), up \(1\)” gives \(y=(x-2)^2+1\).
- Map the key points. Apply \((x,y)\mapsto(x+h,\,y+k)\) to intercepts and turning points to locate the image.
- Shift asymptotes, domain and range. Move a vertical asymptote and the domain by \(h\); move a horizontal asymptote and the range by \(k\).
- Sketch and check. Plot the mapped points and translated asymptotes, then confirm the shape matches the original curve.
| \(y\) | \(=\) | \(f(x)+3\) |
| \(=\) | \(x^{2}+3\) |
| \((0,0)\) | \(\mapsto\) | \((0,\,3)\) |
| \(y\) | \(=\) | \(f(x-4)\) |
| \(=\) | \((x-4)^{3}\) |
| \((1,1)\) | \(\mapsto\) | \((5,\,1)\) |
| \((5-4)^{3}\) | \(=\) | \(1^{3}=1\ \checkmark\) |
| \(y\) | \(=\) | \((x-(-3))^{2}+(-2)\) |
| \(=\) | \((x+3)^{2}-2\) |
| \((0,0)\) | \(\mapsto\) | \((-3,\,-2)\) |
| domain | \(:\) | \(x\in\mathbb{R}\) |
| range | \(:\) | \(y\ge -2\) |
| \(y\) | \(=\) | \(\dfrac{1}{x-2}+1\) |
| \(x=0\) | \(\to\) | \(x=2\) |
| \(y=0\) | \(\to\) | \(y=1\) |
| domain | \(:\) | \(x\ne 2\) |
| range | \(:\) | \(y\ne 1\) |
Common pitfalls
Frequently asked questions
What is a translation of a graph?
A translation slides a graph horizontally and/or vertically without changing its shape, size or orientation. Every point moves by the same amount, so the image is congruent to the original.
What does y = f(x - h) do to a graph?
Replacing \(x\) with \(x-h\) translates the graph horizontally by \(h\): right when \(h>0\), left when \(h<0\). So \(y=f(x-3)\) moves it \(3\) units right.
What does y = f(x) + k do to a graph?
Adding \(k\) translates the graph vertically by \(k\): up when \(k>0\), down when \(k<0\). So \(y=f(x)-2\) moves it \(2\) units down.
How do points move under a translation?
For \(y=f(x-h)+k\), every point maps \((x,y)\mapsto(x+h,\,y+k)\). Applying this to intercepts and turning points is the quickest way to draw the image.
How does a translation affect asymptotes, domain and range?
A horizontal shift of \(h\) moves any vertical asymptote and the domain by \(h\); a vertical shift of \(k\) moves any horizontal asymptote and the range by \(k\). So \(x=a\) becomes \(x=a+h\) and \(y=b\) becomes \(y=b+k\).
Why does x - h move the graph to the right?
The curve reaches a given height when \(x-h\) equals the original input, so \(x\) must be \(h\) larger. Because every point needs a larger \(x\), the whole graph shifts \(h\) units right.