Dilations of Linear Functions

Q: How is a dilation different from a translation?

A translation slides the graph without changing its steepness — the lines stay parallel. A dilation multiplies the outputs, which changes the steepness. If the new line isn't parallel to the old one, you're looking at a dilation.

Q: Why doesn't the line move at the origin?

A point on the $x$-axis has output $0$, and $a \cdot 0 = 0$ for any $a$. So any point where the graph crosses the $x$-axis stays fixed under a dilation.

Q: What does a negative $a$ do?

It reflects the graph over the $x$-axis and applies the stretch or compression from $|a|$. For example, $a = -3$ makes the line three times as steep and flips rising to falling.

Q: Does $g(x) = 4f(x)$ change the $y$-intercept?

It multiplies it by $4$. If $f$ crosses the $y$-axis at $2$, then $g$ crosses at $8$. Only an intercept of $0$ stays put — which is why dilations of $f(x) = x$ keep passing through the origin.

A dilation stretches or squashes a graph instead of sliding it. For a function $f$ , the dilation is $g(x) = a \cdot f(x)$ : every output of $f$ gets multiplied by the number $a$ . Points far from the $x$ -axis move farther; points on the $x$ -axis stay put.

For a line, multiplying every output changes the steepness. That gives you a three-part rule worth memorizing: $a$ bigger than $1$ makes the line steeper, $a$ between $0$ and $1$ makes it flatter, and a negative $a$ also flips the line over the $x$ -axis.

What multiplying the output does

Take $f(x) = x$ and dilate it by $a = 2$ to get $g(x) = 2x$ . The point $(1, 1)$ on $f$ becomes $(1, 2)$ on $g$ , and $(3, 3)$ becomes $(3, 6)$ — every $y$ -value doubles while the $x$ -values stay where they were. The slope doubles from $1$ to $2$ , so the line is steeper.

The graph below shows $f(x) = x$ in blue and $g(x) = 2x$ in red. Both pass through the origin, because multiplying an output of $0$ still gives $0$ .

Reading the effect from $a$

Compare $|a|$ to $1$ . If $|a| > 1$ , outputs grow and the line gets steeper — a vertical stretch. If $0 < |a| < 1$ , outputs shrink toward the $x$ -axis and the line gets flatter — a vertical compression.

The sign of $a$ is a separate piece of information. A negative $a$ reflects the graph over the $x$ -axis: a rising line becomes a falling one. So $g(x) = -\dfrac{1}{2}f(x)$ does two things at once — it compresses the line to half as steep and flips it over the $x$ -axis.

Finding $a$ from a graph

Pick an $x$ -value where you can read both graphs cleanly, then divide: $a = \dfrac{g(x)}{f(x)}$ . If $f$ passes through $(1, 1)$ and $g$ passes through $(1, 3)$ , then $a = 3$ . For lines through the origin, comparing slopes works just as well — $a$ is the ratio of the new slope to the old one.

The table below lists $f(x) = x$ and $g(x) = 2x$ together. Dividing any $g$ -output by the matching $f$ -output gives $a = 2$ every time — that constant ratio is the dilation factor.

$x$	f(x) = x	g(x) = 2x
$-2$	$-2$	$-4$
$-1$	$-1$	$-2$
$1$	$1$	$2$
$2$	$2$	$4$
$3$	$3$	$6$

Worked examples

Example 1: a vertical stretch

Let $f(x) = x$ . Find $g(x) = 2f(x)$ and describe the graph.

Multiply the function by

2

g(x) = 2 \cdot x

Simplify

g(x) = 2x

The slope went from

1

2

, so the line is steeper

Answer: $g(x) = 2x$ — steeper than $f$ .

Example 2: a compression with a flip

Let $f(x) = x$ . Find $g(x) = -\dfrac{1}{2}f(x)$ and describe the graph.

Multiply the function by

-\dfrac{1}{2}

g(x) = -\dfrac{1}{2}x

|a| = \dfrac{1}{2} < 1

, so the line is flatter

a < 0

, so the line is also reflected over the

x

-axis

Answer: $g(x) = -\dfrac{1}{2}x$ — flatter and flipped.

Example 3: find $a$ from two lines

The graphs of $f(x) = x$ and $g(x) = a \cdot f(x)$ both pass through the origin. $g$ passes through $(1, 3)$ . Find $a$ .

Evaluate both functions at

x = 1

f(1) = 1,\quad g(1) = 3

Divide the outputs

a = \dfrac{g(1)}{f(1)} = \dfrac{3}{1}

Simplify

a = 3

Answer: $a = 3$

Try one yourself

Common questions

How is a dilation different from a translation?

A translation slides the graph without changing its steepness — the lines stay parallel. A dilation multiplies the outputs, which changes the steepness. If the new line isn't parallel to the old one, you're looking at a dilation.

Why doesn't the line move at the origin?

A point on the $x$ -axis has output $0$ , and $a \cdot 0 = 0$ for any $a$ . So any point where the graph crosses the $x$ -axis stays fixed under a dilation.

What does a negative

a

do?

It reflects the graph over the $x$ -axis and applies the stretch or compression from $|a|$ . For example, $a = -3$ makes the line three times as steep and flips rising to falling.

Does

g(x) = 4f(x)

change the

y

-intercept?

It multiplies it by $4$ . If $f$ crosses the $y$ -axis at $2$ , then $g$ crosses at $8$ . Only an intercept of $0$ stays put — which is why dilations of $f(x) = x$ keep passing through the origin.

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Dilations of Linear Functions

What multiplying the output does

Reading the effect from $a$

Finding $a$ from a graph

Worked examples

Example 1: a vertical stretch

Example 2: a compression with a flip

Example 3: find $a$ from two lines

Try one yourself

Common questions

Keep learning

Vertical Translations of Linear Functions

Horizontal Translations of Linear Functions

Direct Variation

Rate of Change and Slope

Dilations of Linear Functions

What multiplying the output does

Reading the effect from aaa

Finding aaa from a graph

Worked examples

Example 1: a vertical stretch

Example 2: a compression with a flip

Example 3: find aaa from two lines

Try one yourself

Common questions

Keep learning

Vertical Translations of Linear Functions

Horizontal Translations of Linear Functions

Direct Variation

Rate of Change and Slope

Reading the effect from $a$

Finding $a$ from a graph

Example 3: find $a$ from two lines