Horizontal and Vertical Translations Revision Notes for HSC SSCE Mathematics Advanced

Horizontal and Vertical Translations

Introduction

Translations are fundamental transformations in mathematics that allow us to move graphs around the coordinate plane. Understanding how to translate functions is essential for sketching graphs and analysing function behaviour.

After studying this topic, you will be able to:

Describe how replacing $x$ with $x - a$ creates a horizontal translation
Describe how replacing $y$ with $y - b$ (or $y = f(x) + b$ ) creates a vertical translation
Determine the equation of a function after translation
Sketch translated functions and identify effects on key features like vertices and intercepts

What is a translation?

A translation is a transformation that moves every point of a graph by the same distance in the same direction. This could be left or right (horizontal translation), or up or down (vertical translation).

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The shape and size of the graph do not change during a translation. Only its position on the coordinate plane changes.

Horizontal translations

A horizontal translation moves a graph left or right along the $x$ -axis.

For a function $y = f(x)$ , the transformed function $y = f(x - a)$ produces a horizontal translation of $a$ units.

The horizontal translation formula

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Horizontal Translation Formula

$y = f(x - a)$

The value of $a$ determines the direction of movement:

When $a > 0$ : the graph moves right by $a$ units
When $a < 0$ : the graph moves left by $a$ units

chatImportant

This might seem counterintuitive at first. Subtracting a positive value actually moves the graph to the right, whilst subtracting a negative value (which is the same as adding) moves it to the left.

Remember: For horizontal translations, the direction is opposite to what you might expect!

Example of horizontal translation

Consider the basic parabola $f(x) = x^2$ , which has its vertex at the origin $(0, 0)$ .

When we apply the transformation $y = f(x - 2) = (x - 2)^2$ , the parabola shifts 2 units to the right. The vertex moves from $(0, 0)$ to (2, 0).

The key point to remember is that horizontal translations affect the x-coordinates of all points on the graph. The $y$ -coordinates remain unchanged.

Worked example: horizontal translation

lightbulbExample

Worked Example: Horizontal Translation of a Parabola

Given: The function $f(x) = x^2 + 1$ and its graph.

Part a: Finding the equation after translation

Question: Determine the equation after a horizontal translation of 3 units left.

Solution:

A horizontal translation of 3 units to the left means we use the transformation $y = f(x - a)$ where $a = -3$ .

The new function will be $y = f(x + 3)$ .

Starting with the original function:

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

Applying the translation by replacing x with (x + 3):

$f(x) = (x + 3)^2 + 1$

The translated graph has equation f(x) = (x + 3)² + 1.

Part b: Finding the vertex

Question: Determine the vertex of the translated graph.

Solution:

The vertex of the original graph $f(x) = x^2 + 1$ is at $(0, 1)$ .

A horizontal translation shifts the x-coordinate by the translation value, whilst the y-coordinate stays the same.

Since we translated 3 units to the left (which is $-3$ ), we subtract 3 from the $x$ -coordinate:

New vertex $= (0 - 3, 1)$

New vertex = (-3, 1)

We can verify this by substituting $x = -3$ into the translated equation:

f(x) = (x + 3)^2 + 1 f(-3) = (-3 + 3)^2 + 1 f(-3) = 0^2 + 1 f(-3) = 1

This confirms the vertex is at (-3, 1).

Part c: Sketching the graphs

Question: Sketch both the original and translated graphs, showing the vertices.

Solution:

We plot both parabolas, marking their vertices clearly.

The blue parabola on the left is the translated version, shifted 3 units to the left of the original teal parabola.

Vertical translations

A vertical translation moves a graph up or down along the $y$ -axis.

For a function $y = f(x)$ , the transformed function $y = f(x) + b$ produces a vertical translation of $b$ units.

The vertical translation formula

infoNote

Vertical Translation Formula

$y = f(x) + b$

The value of $b$ determines the direction of movement:

When $b > 0$ : the graph moves up by $b$ units
When $b < 0$ : the graph moves down by $b$ units

This is more intuitive than horizontal translations. Adding a positive value moves the graph up, and adding a negative value (subtracting) moves it down.

Example of vertical translation

Consider again the basic parabola $f(x) = x^2$ , with vertex at $(0, 0)$ .

When we apply the transformation $y = f(x) + 3 = x^2 + 3$ , the parabola shifts 3 units upward. The vertex moves from $(0, 0)$ to (0, 3).

Vertical translations affect the y-coordinates of all points on the graph. The $x$ -coordinates remain unchanged.

Worked example: vertical translation

lightbulbExample

Worked Example: Vertical Translation of a Linear Function

Given: The function $f(x) = 2x$

Part a: Finding the equation after translation

Question: Find the equation after a vertical translation of 4 units down.

Solution:

A vertical translation of 4 units down means we use the transformation $y = f(x) + b$ where $b = -4$ .

Starting with the original function:

$f(x) = 2x$

Applying the translation:

$y = f(x) + b$

Substituting $b = -4$ :

$y = f(x) - 4$

Substituting $f(x) = 2x$ :

$y = 2x - 4$

The translated graph has equation y = 2x - 4.

Part b: Sketching both graphs

Question: Sketch both graphs, including the $y$ -intercept.

Solution:

For the original function $y = 2x$ :

The $y$ -intercept is at (0, 0)
The slope is 2

For the translated function $y = 2x - 4$ :

The $y$ -intercept is at (0, -4)
The slope remains 2 (unchanged)

Both lines are parallel because they have the same slope. The translated line is shifted 4 units downward from the original.

Combining horizontal and vertical translations

Functions can undergo both horizontal and vertical translations. When this happens, we apply both transformations:

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Combined Translation Formula

$y = f(x - a) + b$

This represents:

A horizontal translation of $a$ units (right if $a > 0$ , left if $a < 0$ )
A vertical translation of $b$ units (up if $b > 0$ , down if $b < 0$ )

For example, if $f(x) = x^2$ , then $y = (x - 2)^2 + 3$ represents:

A horizontal translation 2 units right (because of $x - 2$ )
A vertical translation 3 units up (because of $+3$ )

The vertex would move from $(0, 0)$ to (2, 3).

Key features and translations

When a graph is translated, different features are affected in specific ways:

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How Translations Affect Key Features

Horizontal translation ( $y = f(x - a)$ ):

Changes the x-coordinates of all points
Affects $x$ -intercepts and vertices
Does not change $y$ -intercepts (usually)

Vertical translation ( $y = f(x) + b$ ):

Changes the y-coordinates of all points
Affects $y$ -intercepts and vertices
Does not change $x$ -intercepts (for most functions)

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Key Points to Remember:

A translation shifts a graph without changing its shape or size
Horizontal translations use the form $y = f(x - a)$ :
- When $a > 0$ , the graph moves right
- When $a < 0$ , the graph moves left
- This affects x-coordinates
Vertical translations use the form $y = f(x) + b$ :
- When $b > 0$ , the graph moves up
- When $b < 0$ , the graph moves down
- This affects y-coordinates
Key features like vertices and intercepts translate with the graph
The counterintuitive part: For horizontal translations, minus moves right and plus moves left!

Horizontal and Vertical Translations (HSC SSCE Mathematics Advanced): Revision Notes