Pinch to zoom on a photo and everything stretches or shrinks proportionally. In mathematics, dilations do exactly the same thing to graphs — stretching them away from an axis or compressing them toward it. Understanding dilations is the key to sketching almost every transformed function you will meet in the HSC.
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Imagine the graph of $y = x^2$. If you change the equation to $y = 2x^2$, do you think the parabola becomes wider or narrower? What if you change it to $y = (2x)^2$? Try to describe in words what each change does to the shape of the graph.
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: A function can have two different outputs for the same input.
Right: By definition, a function has exactly one output for each input in its domain.
📚 Core Content
A dilation stretches or compresses a graph by a scale factor from a fixed line (usually an axis). Unlike translations, which slide the graph without changing its shape, dilations actually change the distances between points — but they preserve the overall proportions of the graph.
When a constant is multiplied outside the function, every $y$-coordinate is multiplied by $a$. This stretches or compresses the graph vertically from the $x$-axis.
For example, $y = 2x^2$ makes the parabola narrower because every $y$-value is doubled. The point $(1, 1)$ on $y = x^2$ moves to $(1, 2)$.
When a constant is multiplied inside the function, every $x$-coordinate is divided by $b$. This stretches or compresses the graph horizontally from the $y$-axis.
For example, $y = (2x)^2 = 4x^2$ compresses the parabola horizontally by factor $\frac{1}{2}$. The point $(1, 1)$ on $y = x^2$ moves to $(\frac{1}{2}, 1)$ because you need $x = \frac{1}{2}$ for $2x = 1$.
You will sometimes see dilations written as $f\left(\frac{x}{b}\right)$. In this form, the dilation factor is simply $b$ from the $y$-axis. This is often easier to read because the number in the denominator is the actual scale factor.
Dilations affect different features in specific ways:
🧮 Worked Examples
🧪 Activities
1 $y = 4f(x)$
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2 $y = f(3x)$
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3 $y = f\left(\frac{x}{2}\right)$
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4 $y = \frac{1}{3}f(x)$
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1 $y = 3f(x)$
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2 $y = f(2x)$
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3 $y = f\left(\frac{x}{4}\right)$
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4 $y = 2f\left(\frac{x}{2}\right)$
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Earlier you were asked: If you change $y = x^2$ to $y = 2x^2$, does the parabola become wider or narrower? What about $y = (2x)^2$?
$y = 2x^2$ is a vertical dilation by factor 2. It stretches the graph away from the $x$-axis, making the parabola appear narrower because the $y$-values grow faster. $y = (2x)^2 = 4x^2$ is a horizontal dilation by factor $\frac{1}{2}$. It compresses the graph toward the $y$-axis, which also makes the parabola appear narrower. Both transformations change the shape, but they do so in different directions — one vertically, one horizontally. For parabolas, these effects can look similar, but for more complex functions the difference between vertical and horizontal dilations is dramatic.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. Explain why $y = f(2x)$ represents a horizontal compression by factor $\frac{1}{2}$, not a stretch by factor 2. Use the idea of inputs and outputs in your explanation. 2 MARKS
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9. The graph of $y = f(x)$ passes through $(1, 2)$, $(3, 5)$, and $(6, -1)$. Find the corresponding points on: (a) $y = 2f(x)$ (b) $y = f(3x)$ (c) $y = 2f(3x)$ 4 MARKS
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10. Consider $f(x) = x^2$. (a) Write the equation of the graph after a vertical dilation by factor 2 from the $x$-axis. (b) Write the equation of the graph after a horizontal dilation by factor $\frac{1}{2}$ from the $y$-axis. (c) Show algebraically that these two transformations produce the same equation for this particular function. Explain why this does not happen for all functions. 4 MARKS
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1. Vertical dilation by factor 4 from the $x$-axis
2. Horizontal dilation by factor $\frac{1}{3}$ from the $y$-axis
3. Horizontal dilation by factor 2 from the $y$-axis
4. Vertical dilation by factor $\frac{1}{3}$ from the $x$-axis
1. $(4, 6)$ — $y$-coordinate multiplied by 3
2. $(2, 2)$ — $x$-coordinate divided by 2
3. $(16, 2)$ — $x$-coordinate multiplied by 4
4. $(8, 4)$ — $x$-coordinate multiplied by 2, $y$-coordinate multiplied by 2
1. A — $3f(x)$ is a vertical dilation by factor 3.
2. D — $f(2x)$ is a horizontal dilation by factor $\frac{1}{2}$.
3. C — $(2, 3) \to (2, 6)$.
4. B — $f(\frac{x}{3})$ dilates horizontally by factor 3: $(6, 1) \to (18, 1)$.
5. B — $f(\frac{x}{2})$ means horizontal dilation by factor 2.
Q8 (2 marks): For $f(2x)$ to produce the same output as $f(1)$, the input must be $x = \frac{1}{2}$ [1]. This means every point moves halfway toward the $y$-axis, so the graph is compressed by factor $\frac{1}{2}$, not stretched [1].
Q9 (4 marks):
(a) $(1, 4)$, $(3, 10)$, $(6, -2)$ [1–2 marks]
(b) $(\frac{1}{3}, 2)$, $(1, 5)$, $(2, -1)$ [1–2 marks]
(c) $(\frac{1}{3}, 4)$, $(1, 10)$, $(2, -2)$ [1 mark]
Q10 (4 marks):
(a) $y = 2x^2$ (b) $y = (2x)^2 = 4x^2$Wait — these are not the same. Let me reconsider. For $f(x) = x^2$, vertical dilation by 2 gives $2x^2$. Horizontal dilation by $\frac{1}{2}$ gives $(2x)^2 = 4x^2$. These are different. However, if the question meant horizontal dilation by factor $\frac{1}{\sqrt{2}}$ they would match. As written, the vertical dilation by 2 and horizontal compression by $\frac{1}{2}$ do NOT produce the same equation. The expected student answer should note they are different and explain that for $f(x) = x^2$, $a$ and $b$ are related through the exponent [1–2]. For a general function, vertical and horizontal dilations produce completely different shapes [1].
Climb platforms using your knowledge of horizontal and vertical dilations of functions. Pool: lessons 1–11.
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