Dilations (HSC SSCE Mathematics Advanced): Revision Notes

Dilations

What is a dilation?

A dilation is a transformation that stretches or compresses a curve in one direction. For example, when you apply a dilation to a circle, it becomes an ellipse. Dilations are an important type of transformation, alongside translations and reflections. By combining these transformations, we can convert complex functions into simpler ones.

Stretching a graph vertically

Understanding vertical dilations

Let's compare two functions:

• $y = x(x - 2)$
• $y = 3x(x - 2)$

When we create a table of values for these functions, we notice something interesting:

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$x(x-2)$	$8$	$3$	$0$	$-1$	$0$	$3$	$8$
$3x(x-2)$	$24$	$9$	$0$	$-3$	$0$	$9$	$24$

We can rewrite $y = 3x(x - 2)$ as $\frac{y}{3} = x(x - 2)$. This shows that the stretching happens when we replace $y$ with $\frac{y}{3}$.

The $x$-axis is the axis of dilation. This means:

• Points on the $x$-axis stay in place
• All other points triple their distance from the $x$-axis

Stretching a graph horizontally

Understanding horizontal dilations

Similar to vertical stretching, we can stretch the graph of $y = x(x - 2)$ horizontally away from the $y$-axis by a factor of 3. To do this, we replace $x$ with $\frac{x}{3}$:

$y = \frac{x}{3}\left(\frac{x}{3} - 2\right) = \frac{1}{9}x(x - 6)$

Let's compare the original and new functions using tables:

Original function:

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$x(x-2)$	$8$	$3$	$0$	$-1$	$0$	$3$	$8$

New function:

$x$	$-6$	$-3$	$0$	$3$	$6$	$9$	$12$
$\frac{x}{3}(\frac{x}{3}-2)$	$8$	$3$	$0$	$-1$	$0$	$3$	$8$

The $y$-axis is the axis of dilation for horizontal dilations. This means:

• The point $(0, 0)$ on the $y$-axis doesn't move
• All other points triple their distance from the $y$-axis

Example: Creating an ellipse from a circle

Enlargements

When dilations don't preserve shape

Usually, dilating a figure changes its shape. The figure is no longer similar to the original.

For example, consider an equilateral triangle $ABC$ with its base on the $x$-axis:

• A horizontal dilation with factor 2 creates a squat (wide and short) isosceles triangle $A'BC'$
• A vertical dilation with factor 3 creates a skinny (narrow and tall) isosceles triangle $A''BC$

Creating similar figures with enlargements

However, if we apply two dilations with the same factor (one horizontal and one vertical), the result is similar to the original figure. This combined transformation is called an enlargement.

For instance, applying both a horizontal dilation with factor 2 and a vertical dilation with factor 2 to triangle $ABC$ produces triangle $PBQ$, which is also equilateral.

The factor 2 is called the enlargement factor or similarity factor.

Example: Enlarging a circle

Stretching with a fractional or negative factor

Fractional factors create compressions

When the dilation factor is between 0 and 1, the graph is compressed (squashed) rather than stretched.

For example, applying a vertical dilation with factor $\frac{1}{2}$ to the parabola $y = x^2 + 2$ gives:

$\frac{y}{\frac{1}{2}} = x^2 + 2$

Which simplifies to:

$y = \frac{1}{2}x^2 + 1$

This parabola is compressed (flatter) compared to the original.

Negative factors include reflections

When the dilation factor is negative, the transformation combines:

• A dilation with the positive value of the factor
• A reflection

For example, applying vertical dilations with factors $-1$ and $-\frac{1}{2}$ to $y = x^2 + 2$ produces:

With factor $-1$: $y = -x^2 - 2$

With factor $-\frac{1}{2}$: $y = -\frac{1}{2}x^2 - 1$

This is the reflection in the $x$-axis of the compressed parabola.

Example: Multiple dilations of a parabola

Remember!