Dilations (HSC SSCE Mathematics Advanced): Revision Notes

Dilations

What are dilations?

A dilation is a transformation that changes the size of a function's graph by stretching or compressing it. This transformation can occur in the horizontal direction (along the x-axis), in the vertical direction (along the y-axis), or in both directions simultaneously. Unlike translations that shift a graph's position, dilations change the graph's scale while keeping key features like the general shape intact.

Horizontal dilations

A horizontal dilation stretches or compresses a graph in the x-direction, away from the y-axis. This means that the x-coordinates of all points on the graph are scaled by a certain factor, whilst the y-coordinates remain unchanged.

How horizontal dilations work

For any function $y = f(x)$, we can create a horizontal dilation by replacing $x$ with $\frac{x}{k}$ in the function:

$y = f\left(\frac{x}{k}\right)$

This transformation has the following effects:

• When k > 1: The graph stretches horizontally. Each point $(x, y)$ on the original graph moves to the point $(kx, y)$ on the dilated graph. The graph becomes wider.

• When 0 < k < 1: The graph compresses horizontally. The graph becomes narrower.

Vertical dilations

A vertical dilation stretches or compresses a graph in the y-direction, away from the x-axis. This means that the y-coordinates of all points on the graph are scaled by a certain factor, whilst the x-coordinates remain unchanged.

How vertical dilations work

For any function $y = f(x)$, we can create a vertical dilation by multiplying the entire function by a factor $l$:

$y = lf(x)$

This transformation has the following effects:

• When l > 1: The graph stretches vertically. Each point $(x, y)$ on the original graph moves to the point $(x, ly)$ on the dilated graph. The graph becomes taller or steeper.

• When 0 < l < 1: The graph compresses vertically. The graph becomes flatter or less steep.

Worked example: Horizontal and vertical dilations

Let's examine how dilations affect the parabola $y = x^2$.

Part a: Horizontal dilation by a factor of 3

Part b: Vertical dilation by a factor of 0.5

Part c: Describing the effects

Horizontal dilation ($k = 3$): This stretches each point on the parabola horizontally by a factor of 3, making the parabola 3 times as wide. The point $(1, 1)$ moves outward to $(3, 1)$.

Vertical dilation ($l = 0.5$): This compresses each point on the parabola vertically by a factor of 0.5. Each y-value is halved, making the graph appear flatter. The point $(1, 1)$ on the original parabola moves to $(1, 0.5)$ after the dilation.

Key ideas

A horizontal dilation by factor $k$ is given by $y = f\left(\frac{x}{k}\right)$. It stretches from the y-axis. When k > 1, the graph stretches (becomes wider). When 0 < k < 1, the graph compresses (becomes narrower).

A vertical dilation by factor $l$ is given by $y = lf(x)$. It stretches from the x-axis. When l > 1, the graph stretches (becomes taller). When 0 < l < 1, the graph compresses (becomes flatter).

Combined dilations

A combined dilation applies both horizontal and vertical dilations simultaneously, scaling the entire graph. For a function $y = f(x)$, we replace $x$ with $\frac{x}{k}$ and multiply by $y$ with $\frac{y}{k}$. This gives:

$y = kf\left(\frac{x}{k}\right)$

This formula shows a geometric enlargement or reduction by a scale factor of $k$:

• When k > 1: The graph enlarges. Every point $(x, y)$ maps to $(kx, ky)$, making the graph larger overall.

• When 0 < k < 1: The graph reduces. The graph becomes smaller overall.

Example of geometric enlargement

Consider the function $f(x) = x^2$ over the domain $-1 < x < 1$.

When we apply a geometric enlargement by factor 2, each point $(x, y)$ on the graph maps to $(2x, 2y)$. This scales both the x and y values uniformly.

Worked example: Combined dilation

Let's examine how a combined dilation affects the cubic function $y = x^3$.

Key ideas

A combined dilation by factor $y = kf\left(\frac{x}{k}\right)$ enlarges the graph if k > 1 or reduces it if 0 < k < 1. The dilation scales both axes uniformly from the origin. When sketching, remember that both x and y coordinates are scaled by the same factor, confirming the visual scaling effect.

Remember!