Dilations of Functions
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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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.
Key insight: Vertical dilations affect $y$-coordinates directly. Horizontal dilations affect $x$-coordinates inversely: $f(bx)$ divides $x$-coordinates by $b$.
Key facts
- $af(x)$ dilates vertically by factor $a$
- $f(bx)$ dilates horizontally by factor $\frac{1}{b}$
- How dilations affect coordinates of key points
Concepts
- Why horizontal dilations are counter-intuitive (factor is $\frac{1}{b}$)
- The difference between stretching and compressing
- How dilations affect domain and range
Skills
- Sketch dilated graphs from their equations
- Write the equation of a dilated graph
- Determine new coordinates after dilation
- Combine dilations with translations and reflections
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.
Vertical Dilations: $y = af(x)$
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.
- If $a > 1$, the graph is stretched away from the $x$-axis
- If $0 < a < 1$, the graph is compressed toward the $x$-axis
- If $a < 0$, there is also a reflection in 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 $(1, 2)$.
Horizontal Dilations: $y = f(bx)$
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.
- If $b > 1$, the graph is compressed toward the $y$-axis (dilation factor $\frac{1}{b}$)
- If $0 < b < 1$, the graph is stretched away from the $y$-axis (dilation factor $\frac{1}{b} > 1$)
- If $b < 0$, there is also a reflection in 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 $2x = 1$.
Alternative Form: $y = f\!\left(\frac{x}{b}\right)$
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.
- $y = f\!\left(\frac{x}{3}\right)$ is a horizontal dilation by factor 3
- $y = f\!\left(\frac{x}{\frac{1}{2}}\right) = f(2x)$ is a horizontal dilation by factor $\frac{1}{2}$
Vertical dilation: $y = af(x)$ — factor $a$ from the $x$-axis — $(x,y) \to (x, ay)$ — $a > 1$: stretch; $0 < a < 1$: compress; Horizontal dilation: $y = f(bx)$ — factor $\tfrac{1}{b}$ from the $y$-axis — $(x,y)...
Pause — copy both dilation rules with their coordinate maps: $y = af(x)$ maps $(x,y) \to (x,ay)$; $y = f(bx)$ maps $(x,y) \to (x/b, y)$ into your book.
Did you get this? True or false: $y = f(3x)$ represents a horizontal dilation by factor 3 from the $y$-axis.
Quick check: The point $(4, 2)$ lies on $y = f(x)$. What is the corresponding point on $y = 2f(x)$?
We just saw that $y = af(x)$ scales $y$-values by $a$ and $y = f(bx)$ scales $x$-values by $\frac{1}{b}$. That raises a question: if every point shifts, how do specific features like intercepts, turning points, and asymptotes transform? This card answers it → vertical dilation leaves $x$-intercepts unchanged; horizontal dilation leaves the $y$-intercept unchanged but moves $x$-intercepts.
Dilations affect different features in specific ways:
- Vertical dilation $af(x)$:
- $x$-intercepts stay the same (where $y = 0$)
- $y$-intercept is multiplied by $a$
- Range is scaled by factor $a$
- Domain is unchanged
- Horizontal dilation $f(bx)$:
- $y$-intercept stays the same (where $x = 0$)
- $x$-intercepts are divided by $b$
- Domain is scaled by factor $\frac{1}{b}$
- Range is unchanged
Vertical dilation: $x$-intercepts unchanged; $y$-intercept multiplied by $a$; range scaled by $a$; domain unchanged; Horizontal dilation: $y$-intercept unchanged; $x$-intercepts divided by $b$; domain scaled by $\tfrac{1}{b}$; range unchanged
Pause — copy the two-column feature table: vertical dilation ($x$-intercepts fixed, $y$-intercept $\times a$) vs horizontal dilation ($y$-intercept fixed, $x$-intercepts $\div b$) into your book.
Quick check: If $y = f(x)$ has a $y$-intercept at $(0, 3)$, what is the $y$-intercept of $y = f(5x)$?
Common mistakes · the 4 traps that cost marks
Thinking $f(bx)$ dilates by factor $b$ horizontally
This is the single most common error with dilations. $f(2x)$ does not stretch by factor 2 — it compresses by factor $\frac{1}{2}$. The number inside the bracket divides the $x$-coordinates, which makes the graph narrower, not wider.
✓ Fix: For $f(bx)$, always write the dilation factor as $\frac{1}{b}$. For $f(\frac{x}{b})$, the factor is $b$.
Confusing vertical and horizontal dilations
Students often describe $y = 2f(x)$ as a horizontal stretch and $y = f(2x)$ as a vertical stretch. The location of the coefficient determines which axis the dilation is from.
✓ Fix: Outside = vertical ($y$-direction). Inside = horizontal ($x$-direction).
Forgetting that dilations preserve the sign of intercepts
A vertical dilation by factor 2 will double a $y$-intercept, but it will not change its sign (unless the dilation factor is negative). Similarly, horizontal dilations do not change $y$-intercepts because $x = 0$ maps to $0$ regardless of the scale factor.
✓ Fix: $y$-intercepts are unaffected by horizontal dilations. $x$-intercepts are unaffected by vertical dilations.
Describing $f(2x)$ as "narrower" without specifying the axis
In exam questions, vague descriptions like "the graph is narrower" may not score full marks. You must specify whether the dilation is from the $x$-axis or the $y$-axis.
✓ Fix: Always say "vertical dilation by factor ... from the $x$-axis" or "horizontal dilation by factor ... from the $y$-axis."
Worked examples · reveal as you go
Describe the transformation that maps $y = f(x)$ to $y = 3f(x)$.
Describe the transformation that maps $y = f(x)$ to $y = f(2x)$.
The graph of $y = f(x)$ passes through $(2, 3)$ and $(6, -1)$. Find the corresponding points on $y = 2f\!\left(\frac{x}{3}\right)$.
Fill the blanks: drag each token into the matching blank.
For $y = af(x)$, the coefficient $a$ is ___ the function — vertical dilation. For $y = f(bx)$, the coefficient $b$ is ___ — horizontal dilation by factor ___. For $y = f(\frac{x}{b})$, the horizontal dilation factor is ___.
Activity 1 — Describe the dilation
For each equation, describe the dilation from $y = f(x)$. State whether it is vertical or horizontal, and give the dilation factor.
$y = 4f(x)$
$y = f(3x)$
$y = f\!\left(\frac{x}{2}\right)$
$y = \frac{1}{3}f(x)$
Odd one out: Which of these is a horizontal dilation?
Quick-fire practice · 5 reps +2 XP per reveal
Describe the dilation that maps $y = f(x)$ to $y = \frac{1}{2}f(x)$.
If $(6, 4)$ lies on $y = f(x)$, find the corresponding point on $y = f(3x)$.
The $y$-intercept of $y = f(x)$ is $(0, -2)$. What is the $y$-intercept of $y = 5f(x)$?
$y = f(x)$ has an $x$-intercept at $(4, 0)$. What is the $x$-intercept of $y = f\!\left(\frac{x}{2}\right)$?
Explain why $y = f(2x)$ represents a horizontal compression by factor $\frac{1}{2}$, not a stretch by factor 2.
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.
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q8. 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.
Q9. 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) $y = 2f(3x)$
Q10. 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.
📖 Comprehensive answers (click to reveal)
Multiple choice — drill bank
MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.
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.
Activity 1 — Describe the dilation model answers
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
Short answer model answers
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) $y = 2x^2$ [1]
(b) Horizontal dilation by factor $\frac{1}{2}$ means $f(2x) = (2x)^2 = 4x^2$ — wait, this is not the same as $2x^2$.
The expected answer: vertical dilation by 2 gives $2x^2$; horizontal compression by $\frac{1}{2}$ gives $(2x)^2 = 4x^2$. These are different [1]. However, if the student uses $f(\frac{x}{\sqrt{2}})$ then $(\frac{x}{\sqrt{2}})^2 = \frac{x^2}{2} \cdot 2 = x^2$... For full marks, explain that for $f(x) = x^2$, the relationship $af(x) = f(\sqrt{a}x)$ holds because of the even exponent [1–2]. For a general function this equivalence does not hold [1].
Five timed questions on dilations of functions. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
⚔ Enter the arenaClimb platforms, hit checkpoints, and answer dilation questions. Pool: lessons 1–11.
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