Determining Transformations
Learn to determine transformations for Queensland Year 11 Mathematical Methods (QCAA). Working backwards from a base graph to its image, you identify which dilation, reflection and translation were applied.
You will learn to match an image to its transformed form, read the shift from a turning point, endpoint or asymptotes, and find the dilation factor from a known point — recovering the effect of the parameters a, b, h and k.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA), determining a transformation means working backwards: given a base graph \(y=f(x)\) and its image, read off the dilation, reflection and translation. Matching the image to \(y=a\,f(x-h)+k\) gives the values of \(a\), \(h\) and \(k\) and the full sequence of transformations.
To determine a transformation, compare the image rule with the general form \(y=a\,f(x-h)+k\). The dilation/reflection factor \(a\) is the number multiplying the function (negative means a reflection in the \(x\)-axis), \(h\) is the horizontal translation and \(k\) is the vertical translation.
Use the graph's key features to pin the parameters down: the turning point of a parabola gives \((h,\,k)\); the asymptotes of \(y=\dfrac{a}{x-h}+k\) give \(x=h\) and \(y=k\); the endpoint of \(y=a\sqrt{x-h}+k\) gives \((h,\,k)\). The factor \(a\) then comes from one more known point.
Match the image to the general transformed form:
Read the parameters from key features:
How to determine a transformation
- Locate the moved feature: read the new turning point, endpoint or asymptotes to get \(h\) and \(k\).
- Find \(a\): substitute one more known point into \(y=a\,f(x-h)+k\) and solve; a negative \(a\) signals a reflection in the \(x\)-axis.
- State the sequence: list the dilation/reflection first, then the translation \(h\) right and \(k\) up.
Match to \(y=a\,f(x-h)+k\):
| \(a\) | \(=\) | \(1\) |
| \(x-h\) | \(=\) | \(x-4\) |
| \(h\) | \(=\) | \(4\) |
| \(k\) | \(=\) | \(1\) |
Here \(a=1\), so there is no dilation or reflection.
A translation \(4\) units right and \(1\) unit up; \(h=4,\ k=1\).
Match to \(y=a\,f(x-h)+k\):
| \(a\) | \(=\) | \(-2\) |
| \(x-h\) | \(=\) | \(x+1\) |
| \(h\) | \(=\) | \(-1\) |
| \(k\) | \(=\) | \(5\) |
\(a=-2\): a dilation by factor \(2\) from the \(x\)-axis and a reflection in the \(x\)-axis. Then \(h=-1\) is \(1\) left, and \(k=5\) is \(5\) up.
Check the vertex \((0,\,0)\to(-1,\,5)\):
| \(y\) | \(=\) | \(-2(-1+1)^2+5\) |
| \(=\) | \(5\ \checkmark\) |
Reflect in the \(x\)-axis, dilate \(\times 2\), then \(1\) left and \(5\) up.
Match to \(y=\dfrac{a}{x-h}+k\):
| \(a\) | \(=\) | \(3\) |
| \(h\) | \(=\) | \(2\) |
| \(k\) | \(=\) | \(-1\) |
So: a dilation by factor \(3\) from the \(x\)-axis, then \(2\) right and \(1\) down.
Asymptotes — \(x=h\) and \(y=k\):
| \(x\) | \(=\) | \(2\) |
| \(y\) | \(=\) | \(-1\) |
Dilate \(\times 3\), then \(2\) right and \(1\) down; asymptotes \(x=2\) and \(y=-1\).
Match to \(y=a\sqrt{x-h}+k\):
| \(a\) | \(=\) | \(-1\) |
| \(h\) | \(=\) | \(-1\) |
| \(k\) | \(=\) | \(2\) |
\(a=-1\): a reflection in the \(x\)-axis; then \(1\) left and \(2\) up.
Image of \((4,\,2)\) — apply \((x,\,y)\to(x-1,\ -y+2)\):
| \(x'\) | \(=\) | \(4-1=3\) |
| \(y'\) | \(=\) | \(-2+2=0\) |
Reflect in the \(x\)-axis, then \(1\) left and \(2\) up; \((4,\,2)\) maps to \((3,\,0)\).
Common pitfalls
Frequently asked questions
How do I determine the transformation from a base graph to its image?
Match the image rule to \(y=a\,f(x-h)+k\): read \(h\) and \(k\) from the moved turning point, endpoint or asymptotes, then find \(a\) from another known point.
How do I find h and k from a graph?
For a parabola or square-root graph, \((h,\,k)\) is the new turning point or endpoint; for a reciprocal graph the asymptotes \(x=h\) and \(y=k\) give them directly.
How do I know if there is a reflection?
Check the sign of \(a\). If \(a\) is negative the graph is reflected in the \(x\)-axis; the size of \(a\) is the dilation factor.
Why does x + 1 mean a shift to the left?
Because \(x+1=x-(-1)\), so \(h=-1\). The horizontal shift is \(-1\), which is \(1\) unit to the left.
How can I check my transformation is correct?
Track a known point through the mapping \((x,\,y)\to(x+h,\ ay+k)\) and confirm it lands on the image, or substitute an \(x\)-value into both rules.