What does a negative value of a do in y equals a f of x minus h plus k?

A negative a reflects the graph in the x axis as well as dilating it. So y equals minus two f of x flips the graph and stretches it by factor two.

What does a negative value of a do in y equals a f of x minus h plus k?

A negative a reflects the graph in the x axis as well as dilating it. So y equals minus two f of x flips the graph and stretches it by factor two.

Year 11 Methods (Unit 1 & 2) Functions, Relations And Transformations

Combinations Of Transformations

Q: What is the combined transformation rule?

The image of y equals f of x is y equals a times f of x minus h, plus k. This is a dilation or reflection by a, then a translation h right and k up.

Q: Does the order of transformations matter?

Yes, when a dilation or reflection is combined with a translation. For y equals a times f of x minus h plus k you scale by a first, then translate, which is why a point maps to x plus h, a times y plus k.

Q: How do I find the new turning point after a combination?

The turning point of y equals f of x at the origin moves to the point h, k, because the dilation fixes the origin and the translation shifts it by h and k.

Q: How do I track a single point through a combination?

Apply the mapping x, y goes to x plus h, a times y plus k. Shift the x coordinate by h, and scale the y coordinate by a before adding k.

20 practice questions 1 video lesson Theory + worked examples

Master combinations of transformations for Queensland Year 11 Mathematical Methods (QCAA). Real graphs often need several changes at once, applying a dilation or reflection and then a translation to a base graph.

You will learn to build the combined rule from the parameters a, b, h and k, apply the correct order — dilate or reflect first, then translate — and track how the whole graph moves to sketch the curve.

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Theory

In Year 11 Mathematical Methods (QCAA), a combination of transformations applies a dilation or reflection and then a translation to a base graph \(y=f(x)\). The image is \(y=a\,f(x-h)+k\): dilate/reflect by \(a\) first, then translate \(h\) right and \(k\) up. This page shows how to build the combined rule, apply it to a base function, and track a point.

A combination of transformations chains several single moves. The standard combined form is \(y=a\,f(x-h)+k\), where \(a\) is a dilation from the \(x\)-axis (a reflection in the \(x\)-axis as well when \(a\lt 0\)), \(h\) is a horizontal translation and \(k\) is a vertical translation.

The usual order for reading \(y=a\,f(x-h)+k\) is dilation/reflection first, then translation. Under it a point \((x,\,y)\) on \(y=f(x)\) maps to \((x+h,\ ay+k)\): the \(x\)-coordinate is shifted by \(h\), while the \(y\)-coordinate is scaled by \(a\) and then shifted by \(k\).

Scale before you shift. Apply \(a\) to the height first, then add the translations \(h\) and \(k\) — doing the translation first gives a different image.

\(y=x^2\) (navy) \(\to\) \(y=2(x-1)^2+3\) (gold): dilate \(2\), right \(1\), up \(3\).

\(y=x^2\) (navy) \(\to\) \(y=-(x-1)^2+2\) (gold): reflect, right \(1\), up \(2\).

Combined transformation of \(y=f(x)\):

\[y=a\,f(x-h)+k\]

y = a f (x - h) + k

Effect on a point (scale then shift):

\[(x,\,y)\ \longmapsto\ (x+h,\ ay+k)\]

(x, y) \mapsto (x + h, a y + k)

Vertex/endpoint shortcut: for \(y=a\,f(x-h)+k\) the turning point or endpoint of \(y=f(x)\) at \((0,\,0)\) moves straight to \((h,\,k)\).

How to combine transformations

Identify \(a,\ h,\ k\): the dilation/reflection factor \(a\), the horizontal shift \(h\) (right positive) and the vertical shift \(k\) (up positive).
Build the rule: substitute the base function into \(y=a\,f(x-h)+k\) and simplify if asked.
Track the key point: send \((x,\,y)\to(x+h,\ ay+k)\) to locate the new turning point, endpoint or a chosen point.

Example 1 — Describe a combination from the rule

Describe the sequence of transformations taking \(y=x^2\) to \(y=2(x-1)^2+3\), and state the turning point.

Solution

Match to \(y=a\,f(x-h)+k\):

\(a\)	\(=\)	\(2\)
\(h\)	\(=\)	\(1\)
\(k\)	\(=\)	\(3\)

So: a dilation by factor \(2\) from the \(x\)-axis, then a translation \(1\) right and \(3\) up.

Turning point — \((0,\,0)\to(h,\,k)\):

\((0,\,0)\)

\(\mapsto\)

\((1,\,3)\)

Dilation \(\times 2\), then \(1\) right and \(3\) up; turning point \((1,\,3)\).

(1, 3)

Example 2 — Reflection combined with a translation

Apply to \(y=x^2\): a reflection in the \(x\)-axis, then a translation \(1\) unit right and \(2\) units up. Write the image rule.

Solution

Read the parameters:

\(a\)	\(=\)	\(-1\)
\(h\)	\(=\)	\(1\)
\(k\)	\(=\)	\(2\)

Rule — substitute into \(y=a\,f(x-h)+k\):

\(y\)

\(=\)

\(-\,(x-1)^2+2\)

Check the vertex — \((0,\,0)\to(1,\,2)\):

\(y\)	\(=\)	\(-(1-1)^2+2\)
	\(=\)	\(2\ \checkmark\)

Image rule \(y=-(x-1)^2+2\), a maximum turning point at \((1,\,2)\).

y = - (x - 1) ⱼ + 2

Example 3 — Build a rule on a square-root graph

The graph of \(y=\sqrt{x}\) is dilated by factor \(2\) from the \(x\)-axis, then translated \(3\) right and \(1\) down. Write the rule and give the endpoint.

Solution

Parameters:

\(a\)	\(=\)	\(2\)
\(h\)	\(=\)	\(3\)
\(k\)	\(=\)	\(-1\)

Rule — use \(y=a\,f(x-h)+k\):

\(y\)	\(=\)	\(2\sqrt{x-3}+(-1)\)
	\(=\)	\(2\sqrt{x-3}-1\)

Endpoint — \((0,\,0)\to(h,\,k)\):

\((0,\,0)\)

\(\mapsto\)

\((3,\,-1)\)

Image rule \(y=2\sqrt{x-3}-1\); the endpoint is \((3,\,-1)\).

y = 2 \sqrt{x - 3} - 1

Example 4 — Track a point through a combination

Under \(y=\tfrac{1}{2}(x+2)^2-3\), the image of \(y=x^2\), find where the point \((2,\,4)\) lands.

Solution

Read the parameters (\(x+2=x-(-2)\)):

\(a\)	\(=\)	\(\tfrac{1}{2}\)
\(h\)	\(=\)	\(-2\)
\(k\)	\(=\)	\(-3\)

Apply \((x,\,y)\to(x+h,\ ay+k)\):

\(x'\)	\(=\)	\(2+(-2)\)
	\(=\)	\(0\)
\(y'\)	\(=\)	\(\tfrac{1}{2}\times 4+(-3)\)
	\(=\)	\(2-3\)
	\(=\)	\(-1\)

The point \((2,\,4)\) maps to \((0,\,-1)\).

(0, - 1)

Common pitfalls

Doing the translation before the dilation. For \(y=a\,f(x-h)+k\) the height is scaled by \(a\) first, then shifted by \(k\); reversing the order moves the graph to the wrong place.

Scaling the \(k\) as well. In \((x,\,y)\to(x+h,\ ay+k)\) only the original \(y\) is multiplied by \(a\); the \(k\) is added afterwards and is not scaled.

Sign of \(h\) again. \(y=a\,f(x+2)+k\) is a shift \(2\) units left because \(x+2=x-(-2)\), so \(h=-2\).

Frequently asked questions

What is the combined transformation rule?

The image of \(y=f(x)\) is \(y=a\,f(x-h)+k\): a dilation/reflection by \(a\), then a translation \(h\) right and \(k\) up.

Does the order of transformations matter?

Yes when a dilation or reflection is combined with a translation. For \(y=a\,f(x-h)+k\) you scale by \(a\) first, then translate, which is why a point maps to \((x+h,\ ay+k)\).

How do I find the new turning point after a combination?

The turning point of \(y=f(x)\) at \((0,\,0)\) moves to \((h,\,k)\), because the dilation fixes the origin and the translation shifts it by \((h,\,k)\).

What does a negative value of a do in y = a f(x - h) + k?

A negative \(a\) reflects the graph in the \(x\)-axis as well as dilating it, so \(y=-2f(x)\) flips the graph and stretches it by factor \(2\).

How do I track a single point through a combination?

Apply \((x,\,y)\to(x+h,\ ay+k)\): shift the \(x\)-coordinate by \(h\), and scale the \(y\)-coordinate by \(a\) before adding \(k\).

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