AP®MathsCollege BoardPrecalculusRevision NotesPolynomial & Rational FunctionsTransformations of FunctionsVertical & Horizontal Dilations

Vertical & Horizontal Dilations (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Vertical dilations

What are graph dilations?

When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
With a dilation all the points on the graph are moved towards or away from either the x- or the y-axis by a constant scale factor

Two graphs showing a sine wave function with vertical and horizontal dilations. — **Dilations of a graph**

What is a vertical dilation?

If you start with a function $f (x)$ and multiply the output by a constant $a$ (where $a \neq 0$ )
- the new function $g (x) = a f (x)$ is called a multiplicative transformation of $f$
The effect on the graph is a vertical dilation (stretch or compression)
- every point on the graph is moved farther from or closer to the $x$ -axis
  - If $| a | > 1$ , the graph is stretched vertically (pulled away from the $x$ -axis)
  - If $0 < | a | < 1$ , the graph is compressed vertically (pushed toward the $x$ -axis)
- If $a < 0$ , the graph is also reflected over the $x$ -axis (i.e. flipped upside down around the $x$ -axis), in addition to the stretch or compression
The dilation factor is $| a |$
- This tells you how much the graph is stretched or compressed
- Points on the graph after the dilation are $| a |$ times as far from the $x$ -axis as they were before the dilation
E.g. for a function $f (x)$
- $g (x) = \frac{1}{3} f (x)$ stretches the graph vertically by a factor of $\frac{1}{3}$
  - the $y$ -coordinates of all points are multiplied by $\frac{1}{3}$ (while $x$ -coordinates stay the same)
- $h (x) = - 2 f (x)$ stretches the graph by a factor of 2 and reflects it over the $x$ -axis
  - the $y$ -coordinates of all points are multiplied by $- 2$ (while $x$ -coordinates stay the same)

Graph transformation of y=f(x) to y=1/3f(x), showing a vertical stretch with scale factor 1/3, affecting y-coordinates while x-coordinates remain unchanged. — **Example of a vertical dilation**

How does a vertical dilation affect key features?

Every output value is multiplied by $a$ , so
- $y$ -intercept
  - multiplied by $a$
- Local maxima and minima
  - their $y$ -values are multiplied by $a$
    - and if $a < 0$ , maxima become minima and vice versa
- Horizontal asymptotes
  - the $y$ -value defining the asymptote is multiplied by $a$
Features that depend only on $x$ -values are not affected
- Zeros ( $x$ -intercepts) stay in the same place
  - since $a \cdot 0 = 0$
- Vertical asymptotes stay in the same place
- Intervals of increase/decrease stay the same
  - unless $a < 0$ , which reverses them

How do I find the analytical form of a vertical dilation?

$a f (x)$ means 'multiply the entire function $f$ by $a$ '
- So if you have an expression for $f (x)$ , simply multiply it by $a$ to get the analytical form of the transformed function
- Simplify if necessary
E.g. if $f (x) = x^{2} - 3 x + 4$
- Then to find $g (x) = 2 f (x)$
  - $g (x) = 2 (x^{2} - 3 x + 4) = 2 x^{2} - 6 x + 8$

Horizontal dilations

What is a horizontal dilation?

If you start with a function $f (x)$ and multiply the input by a constant $b$ (where $b \neq 0$ ) before the function is applied
- the new function $g (x) = f (b x)$ is also called a multiplicative transformation of $f$
  - i.e., vertical and horizontal dilations are both types of multiplicative transformation
The effect on the graph is a horizontal dilation (stretch or compression)
- every point on the graph is moved farther from or closer to the $y$ -axis
  - If $| b | > 1$ , the graph is compressed horizontally (pushed toward the $y$ -axis) by a factor of $\frac{1}{| b |}$
  - If $0 < | b | < 1$ , the graph is stretched horizontally (pulled away from the $y$ -axis) by a factor of $\frac{1}{| b |}$
- If $b < 0$ , the graph is also reflected over the $y$ -axis (i.e. flipped left-to-right around the $y$ -axis), in addition to the stretch or compression

The dilation factor is $\frac{1}{| b |}$
- This tells you how much the graph is stretched or compressed
- Points on the graph after the dilation are $\frac{1}{| b |}$ times as far from the $y$ -axis as they were before the dilation
Note the counterintuitive behavior
- the dilation factor is $\frac{1}{| b |}$ , not $| b |$
  - Multiplying the input by a number greater than 1 actually compresses the graph, rather than stretching it
E.g. for a function $f (x)$
- $g (x) = f (2 x)$ compresses the graph horizontally by a factor of $\frac{1}{2}$
  - the $x$ -coordinates of all points are multiplied by $\frac{1}{2}$ (while $y$ -coordinates stay the same)
- $h (x) = f (- \frac{1}{3} x)$ stretches the graph horizontally by a factor of 3 and reflects it over the $y$ -axis
  - the $x$ -coordinates of all points are multiplied by $- 3$ (while $y$ -coordinates stay the same)

Graph of y=f(x) is stretched horizontally, forming y=f(2x). X-coordinates change; y-coordinates remain. Points on y-axis are unaffected. — **Example of a horizontal dilation**

How does a horizontal dilation affect key features?

Every $x$ -value on the graph is multiplied by $\frac{1}{b}$ , so features that depend on $x$ -values are affected:
- Zeros ( $x$ -intercepts)
  - their $x$ -values are multiplied by $\frac{1}{b}$
- Local maxima and minima
  - their $x$ -values are multiplied by $\frac{1}{b}$ , but the $y$ -values stay the same
- Vertical asymptotes
  - their $x$ -values are multiplied by $\frac{1}{b}$
Features that depend only on $y$ -values are not affected
- $y$ -intercept stays the same
  - since $f (b \cdot 0) = f (0)$
- Horizontal asymptotes stay at the same $y$ -value

A graph showing transformations of y=f(x) with vertical stretch 2 affecting y, moving horizontal asymptote, and horizontal stretch 3 affecting x, moving vertical asymptote. — **Effects of vertical and horizontal dilations**

How do I find the analytical form of a horizontal dilation?

$f (b x)$ means 'replace $x$ everywhere in $f (x)$ with $b x$ '
- So if you have an expression for $f (x)$ , simply put $b x$ everywhere that $x$ is to get the analytical form of the transformed function
- Simplify if necessary
E.g. if $f (x) = x^{2} - 3 x + 4$
- Then to find $g (x) = f (2 x)$
  - $g (x) = {(2 x)}^{2} - 3 (2 x) + 4 = 4 x^{2} - 6 x + 4$

How can you tell a vertical and horizontal dilation apart?

Vertical dilation $g (x) = a f (x)$
- the constant $a$ multiplies the output
  - it appears outside the function
Horizontal dilation $g (x) = f (b x)$
- the constant $b$ multiplies the input
  - it appears inside the function
A common mistake is to confuse the two
- Just like with translations, the key question is:
  - Is the constant being applied to the input or the output?

Examiner Tips and Tricks

A common error on the exam is forgetting that horizontal dilation by $b$ compresses by a factor of $\frac{1}{| b |}$ (the reciprocal), not $| b |$ .

Another common error is forgetting that a negative value of $a$ or $b$ in a dilation also causes a reflection.

Worked Example

The function $f$ is given by $f (x) = 2 x^{2} - 5 x + 3$ . The graph of which of the following functions is the image of the graph of $f$ after a vertical dilation of the graph of $f$ by a factor of 3?

(A) $p (x) = 2 x^{2} - 5 x + 6$ , because this is an additive transformation of $f$ that results from adding 3 to $f (x)$ .

(B) $q (x) = 2 x^{2} + 7 x + 6$ , because this is an additive transformation of $f$ that results from adding 3 to each input value $x$ .

(C) $r (x) = 6 x^{2} - 15 x + 9$ , because this is a multiplicative transformation of $f$ that results from multiplying $f (x)$ by 3.

(D) $s (x) = 18 x^{2} - 15 x + 3$ , because this is a multiplicative transformation of $f$ that results from multiplying each input value $x$ by 3.

Answer:

Additive transformations cause translations, not dilations

So you can rule out (A), which is a vertical translation up by 3 units
and also (B), which is a horizontal translation to the left by 3 units

Multiplying each input value $x$ by 3 gives

$\begin{array}{rcl} 2 {(3 x)}^{2} - 5 (3 x) + 3 & = & 2 \times 9 x^{2} - 15 x + 3 \\ = & 18 x^{2} - 15 x + 3 \end{array}$

That is function $s$ in option (D)
But multiplying the input value by 3 causes a horizontal dilation by a factor of $\frac{1}{3}$

Multiplying $f (x)$ by 3 gives

$3 (2 x^{2} - 5 x + 3) = 6 x^{2} - 15 x + 9$

That is function $r$ in option (C)
And indeed multiplying the function by 3 causes a vertical dilation by a factor of 3

(C) $r (x) = 6 x^{2} - 15 x + 9$ , because this is a multiplicative transformation
of $f$ that results from multiplying $f (x)$ by 3

Examiner Tips and Tricks

On the exam, incorrect answers in multiple choice questions often include an option that applies the constant to the wrong part of the function.

E.g. multiplying inputs when the question asks for a vertical dilation, or vice versa

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Vertical & Horizontal Dilations (College Board AP® Precalculus): Revision Note

Vertical dilations

What are graph dilations?

What is a vertical dilation?

How does a vertical dilation affect key features?

How do I find the analytical form of a vertical dilation?

Horizontal dilations

What is a horizontal dilation?

How does a horizontal dilation affect key features?

How do I find the analytical form of a horizontal dilation?

How can you tell a vertical and horizontal dilation apart?

Examiner Tips and Tricks

Worked Example

Examiner Tips and Tricks

Unlock more, it's free!

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Author: Roger B

Reviewer: Mark Curtis