Dilations And Reflections
Understand dilations and reflections for Queensland Year 11 Mathematical Methods (QCAA). A dilation stretches or compresses a graph towards or away from an axis, while a reflection flips it over an axis.
You will learn to apply vertical and horizontal dilations, reflect a graph in the x-axis or the y-axis, and see the effect of the parameters a and b on key points, domain and range.
Every question with a fully worked solution.
- Dilations And Reflections - Video - Dilations and Reflections Watch
Theory
In Year 11 Mathematical Methods (QCAA), a dilation stretches or compresses a graph and a reflection flips it over an axis. A vertical dilation gives \(y=a\,f(x)\), a reflection in the \(x\)-axis gives \(y=-f(x)\), and a reflection in the \(y\)-axis gives \(y=f(-x)\). This page shows how each one acts on the key points of a graph.
A dilation from the \(x\)-axis by factor \(a\) multiplies every \(y\)-coordinate by \(a\), giving the image \(y=a\,f(x)\); the point \((x,\,y)\) maps to \((x,\,ay)\). If \(a\gt 1\) the graph is stretched vertically; if \(0\lt a\lt 1\) it is compressed.
A reflection in the \(x\)-axis is the special case \(a=-1\): \(y=-f(x)\), so \((x,\,y)\to(x,\,-y)\). A reflection in the \(y\)-axis replaces \(x\) with \(-x\): \(y=f(-x)\), so \((x,\,y)\to(-x,\,y)\). A dilation from the \(y\)-axis by factor \(b\) stretches horizontally, giving \(y=f\!\left(\dfrac{x}{b}\right)\) and \((x,\,y)\to(bx,\,y)\).
Dilation from the \(x\)-axis by factor \(a\) (vertical):
Reflection in the \(x\)-axis, then in the \(y\)-axis:
Dilation from the \(y\)-axis by factor \(b\) (horizontal):
How to dilate or reflect a graph
- Name the transformation: vertical dilation \(y=a\,f(x)\), horizontal dilation \(y=f(x/b)\), or reflection \(y=-f(x)\) or \(y=f(-x)\).
- Write the rule: a vertical change multiplies the whole function; a horizontal change replaces \(x\).
- Move the key points: apply \((x,\,y)\to(x,\,ay)\), \((x,\,-y)\), \((-x,\,y)\) or \((bx,\,y)\) to the turning point, endpoint or intercepts.
Rule — multiply the function by \(a=3\):
| \(y\) | \(=\) | \(a\,f(x)\) |
| \(=\) | \(3x^2\) |
Point — map \((x,\,y)\to(x,\,3y)\):
| \((1,\,1)\) | \(\mapsto\) | \((1,\ 3\times 1)\) |
| \(=\) | \((1,\,3)\) |
Image rule \(y=3x^2\); the point \((1,\,1)\) maps to \((1,\,3)\).
Rule — take the negative of the function:
| \(y\) | \(=\) | \(-f(x)\) |
| \(=\) | \(-x^2\) |
Point — map \((x,\,y)\to(x,\,-y)\):
| \((2,\,4)\) | \(\mapsto\) | \((2,\,-4)\) |
Image rule \(y=-x^2\); the point \((2,\,4)\) maps to \((2,\,-4)\).
Rule — replace \(x\) with \(-x\):
| \(y\) | \(=\) | \(f(-x)\) |
| \(=\) | \(\sqrt{-x}\) |
Domain — need \(-x\ge 0\):
| \(-x\) | \(\ge\) | \(0\) |
| \(x\) | \(\le\) | \(0\) |
Point — map \((x,\,y)\to(-x,\,y)\):
| \((4,\,2)\) | \(\mapsto\) | \((-4,\,2)\) |
Image rule \(y=\sqrt{-x}\) with domain \(x\le 0\); the point \((4,\,2)\) maps to \((-4,\,2)\).
Rule — replace \(x\) with \(\dfrac{x}{2}\):
| \(y\) | \(=\) | \(f\!\left(\dfrac{x}{2}\right)\) |
| \(=\) | \(\left(\dfrac{x}{2}\right)^{2}\) | |
| \(=\) | \(\dfrac{x^2}{4}\) |
Point — map \((x,\,y)\to(2x,\,y)\):
| \((2,\,4)\) | \(\mapsto\) | \((4,\,4)\) |
Image rule \(y=\dfrac{x^2}{4}\); the point \((2,\,4)\) maps to \((4,\,4)\).
Common pitfalls
Frequently asked questions
What is the difference between a dilation and a reflection?
A dilation stretches or compresses the graph (\(y=a\,f(x)\) or \(y=f(x/b)\)); a reflection flips it over an axis (\(y=-f(x)\) or \(y=f(-x)\)).
How do I write a vertical dilation by factor a?
Multiply the whole function by \(a\): the image of \(y=f(x)\) is \(y=a\,f(x)\), and each point \((x,\,y)\) maps to \((x,\,ay)\).
What does y = f(-x) do to a graph?
It reflects the graph in the \(y\)-axis, so every point \((x,\,y)\) maps to \((-x,\,y)\); left and right are swapped.
Does a reflection or dilation move the origin?
For \(y=a\,f(x)\), \(y=-f(x)\) and \(y=f(-x)\) the origin stays fixed, because multiplying or negating \(0\) still gives \(0\).
How does a dilation change the domain or range?
A vertical dilation stretches the range; a horizontal dilation or a \(y\)-axis reflection changes the domain (for example \(y=\sqrt{-x}\) has domain \(x\le 0\)).