Using transformations to sketch graphs
Theory
Sketching a transformed graph starts from a known base curve — such as \(y=x^{2}\), \(y=\dfrac{1}{x}\), \(y=\sqrt{x}\), \(y=\sin x\) or \(y=2^{x}\) — and applies dilations, reflections and translations written together as \(y=a\,f\big(b(x-h)\big)+k\). You map the base graph's key points and asymptotes to their images, then draw the result with its intercepts, asymptotes, domain and range labelled.
A base graph is a standard curve whose shape you already know: the parabola \(y=x^{2}\), the hyperbola \(y=\dfrac{1}{x}\), the square-root curve \(y=\sqrt{x}\), the exponential \(y=2^{x}\) and the trigonometric curve \(y=\sin x\). A transformation reshapes or repositions this base graph without changing which family it belongs to, so the transformed curve keeps the same essential shape.
There are three families of transformation. A dilation stretches or compresses the graph — a factor \(a\) outside the function stretches it vertically, and a factor \(b\) inside compresses it horizontally. A reflection flips the graph in an axis: \(-f(x)\) flips it in the \(x\)-axis and \(f(-x)\) flips it in the \(y\)-axis. A translation slides the graph, by \(h\) horizontally and \(k\) vertically. Collected together, a transformed standard curve is \(y=a\,f\big(b(x-h)\big)+k\).
To sketch it, you do not plot dozens of points. Instead you map the key features of the base graph — its key points, intercepts, endpoints and any asymptotes — to their new positions, plot those images, and draw the curve through them with the correct shape and end behaviour. Finally you read off and state the domain and range, which the dilations, reflections and translations may have changed.
A transformed standard curve, with base function \(f\):
Each base point \((x,y)\) is sent to its image by the point-mapping rule:
The individual effects — a vertical dilation and \(x\)-axis reflection from \(a\), a horizontal dilation and \(y\)-axis reflection from \(b\), and the translations from \(h,k\):
How to sketch a transformed graph
- Name the base graph. Recognise which standard curve \(f\) you are transforming — \(x^{2}\), \(\dfrac{1}{x}\), \(\sqrt{x}\), \(\sin x\) or \(a^{x}\) — and recall its shape and key features.
- Write it in standard form. Rearrange the rule as \(y=a\,f\big(b(x-h)\big)+k\); factorise the \(b\) out of the bracket so the horizontal shift \(h\) is read correctly.
- Read off the transformations. The dilation factors are \(|a|\) (vertical) and \(\dfrac{1}{|b|}\) (horizontal); negative \(a\) or \(b\) gives a reflection; \(h\) and \(k\) are the translations.
- Map the key features. Send each key point \((x,y)\) to \(\left(\dfrac{x}{b}+h,\,a\,y+k\right)\), and move every asymptote and endpoint to its new position.
- Draw and label. Plot the images, draw the curve with the correct shape and end behaviour, mark the intercepts and asymptotes, and state the domain and range.
| vertex \((0,0)\) | \(\mapsto\) | \((0+2,\;0+1)=(2,1)\) |
| \(y\) | \(=\) | \((0-2)^{2}+1\) |
| \(=\) | \(5\) |
| \(x=0\) | \(\mapsto\) | \(x=1\) (vertical) |
| \(y=0\) | \(\mapsto\) | \(y=2\) (horizontal) |
| \(x=0:\;y\) | \(=\) | \(\dfrac{1}{-1}+2=1\) |
| \(y=0:\;\dfrac{1}{x-1}\) | \(=\) | \(-2\;\Rightarrow\;x=\dfrac{1}{2}\) |
| endpoint \((0,0)\) | \(\mapsto\) | \((0,\;-2(0)+3)=(0,3)\) |
| \((1,1)\) | \(\mapsto\) | \((1,\;-2(1)+3)=(1,1)\) |
| \(3-2\sqrt{x}\) | \(=\) | \(0\) |
| \(\sqrt{x}\) | \(=\) | \(\dfrac{3}{2}\;\Rightarrow\;x=\dfrac{9}{4}\) |
| amplitude | \(=\) | \(|a|=2\) |
| period | \(=\) | \(\dfrac{2\pi}{b}=\dfrac{2\pi}{2}=\pi\) |
| midline | \(=\) | \(y=1\) |
| range | \(=\) | \([\,1-2,\;1+2\,]=[-1,3]\) |
| max | \(=\) | \(\left(\dfrac{\pi}{4},\,3\right)\) |
| min | \(=\) | \(\left(\dfrac{3\pi}{4},\,-1\right)\) |
Common pitfalls
Frequently asked questions
What are the three main types of graph transformation?
Dilations stretch or compress a graph, reflections flip it in the \(x\)- or \(y\)-axis, and translations slide it left, right, up or down. Together they give \(y=a\,f\big(b(x-h)\big)+k\), where \(a,b\) set the dilations and reflections and \(h,k\) the translation.
Which way does the graph of y equals f of (x minus h) move?
To the right by \(h\). A change inside the function acts opposite to its sign, so \(f(x-2)\) shifts \(2\) right and \(f(x+2)\) shifts \(2\) left; adding \(k\) outside moves the graph straight up by \(k\).
How do dilations change a graph?
The outside factor \(a\) multiplies every height, stretching the graph vertically by \(|a|\) (and reflecting in the \(x\)-axis if \(a<0\)). The inside factor \(b\) dilates horizontally by \(\dfrac{1}{|b|}\); for example \(y=\sin(bx)\) has period \(\dfrac{2\pi}{b}\).
How do you sketch a transformed graph step by step?
Write the rule as \(y=a\,f\big(b(x-h)\big)+k\) and read off the dilations, reflections and translations. Then map the base graph's key points, intercepts, endpoints and asymptotes to their images, plot them, draw the curve, and label the domain and range.
How do transformations affect asymptotes?
Asymptotes move with the curve. For \(y=\dfrac{1}{x-1}+2\) the vertical asymptote of \(y=\dfrac{1}{x}\) moves to \(x=1\) and the horizontal one to \(y=2\). Always redraw them in their new positions before sketching the branches.
What is the general form of a transformed function?
It is \(y=a\,f\big(b(x-h)\big)+k\): \(a\) is the vertical dilation (its sign an \(x\)-axis reflection), \(b\) gives the horizontal dilation \(\dfrac{1}{|b|}\) (its sign a \(y\)-axis reflection), and \(h,k\) translate right \(h\) and up \(k\).