Translations (VCE SSCE Mathematical Methods): Revision Notes

Translations

What are transformations?

The Cartesian plane is the set of all ordered pairs of real numbers, written as $\mathbb{R}^2$. In mathematical notation:

$$\mathbb{R}^2 = \{(x, y) : x, y \in \mathbb{R}\}$$

A transformation of the plane is a rule that maps each point in the plane to exactly one new point. Every point becomes the image of another point under the transformation.

Understanding mapping notation

When we translate points, we use mapping notation to show where points move.

Consider translating 2 units right (positive $x$-direction) and 4 units up (positive $y$-direction). We write this as:

$$(x, y) \rightarrow (x + 2, y + 4)$$

This reads as "the point $(x, y)$ maps to $(x + 2, y + 4)$."

We can also express translations using primed coordinates. If $(x', y')$ is the image of point $(x, y)$, then:

$$x' = x + 2 \text{ and } y' = y + 4$$

These equations tell us how to calculate the new coordinates from the original ones.

Translation rules

Translations in positive directions

Translations in negative directions

Applying translations to sketch graphs

When you translate a graph, every point on it moves the same distance in the same direction. This means the shape stays exactly the same, just in a different position.

Translations parallel to the $x$-axis

Let's look at what happens when we translate the basic parabola $y = x^2$ horizontally.

Translation 1 unit to the right:

Each point $(x, y)$ maps to $(x + 1, y)$. The image has equation:

$$y = (x - 1)^2$$

Translation 1 unit to the left:

Each point $(x, y)$ maps to $(x - 1, y)$. The image has equation:

$$y = (x + 1)^2$$

Translations parallel to the $y$-axis

Now let's translate the parabola vertically.

Translation 1 unit up:

Each point $(x, y)$ maps to $(x, y + 1)$. The image has equation:

$$y = x^2 + 1$$

When moving up, we add to the function.

Translation 1 unit down:

Each point $(x, y)$ maps to $(x, y - 1)$. The image has equation:

$$y = x^2 - 1$$

When moving down, we subtract from the function.

General translations of curves

Every translation can be described by combining two components:

• A horizontal translation (parallel to the $x$-axis)
• A vertical translation (parallel to the $y$-axis)

Finding the equation of a translated curve

Consider translating $y = x^2$ by 2 units right and 4 units up. The translation rule is:

$$(x, y) \rightarrow (x + 2, y + 4)$$

This means: $x' = x + 2$ and $y' = y + 4$

To find the image equation, we need to express the original coordinates in terms of the new ones:

$$x = x' - 2 \text{ and } y = y' - 4$$

Substituting into the original equation $y = x^2$:

$$y' - 4 = (x' - 2)^2$$

The image curve has equation: $y - 4 = (x - 2)^2$ (dropping the primes for simplicity).

Important relationship

Worked example

Summary

Remember!