When you study graphs of functions, it’s really important to understand translations. Translations are ways to move the graph of a function without changing its shape. There are two main types: horizontal translations and vertical translations. Let’s take a closer look at how these two types differ.

Horizontal Translations

Horizontal translations are about moving a graph left or right on the x-axis (the horizontal line).

You can think of it like this:

• The general form for a horizontal shift looks like this:

$$f(x) \to f(x - h)$$

In this formula, $h$ tells us how far to move the graph:

• If $h$ is positive: The graph moves to the right.

For example, if you have the function $f(x) = x^2$ and you want to move it 3 units to the right, you write it as:

$$f(x - 3) = (x - 3)^2$$

Now, the top point (called the vertex) goes from (0,0) to (3,0).

• If $h$ is negative: The graph moves to the left.

Using the same function $f(x) = x^2$, if we want to shift it 2 units left, we write it like this:

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

Now, the vertex moves from (0,0) to (-2,0).

Vertical Translations

Vertical translations are about moving the graph up or down on the y-axis (the vertical line).

This is how it looks:

• The general form for a vertical shift is:

$$f(x) \to f(x) + k$$

Here, $k$ shows the direction and how far to move:

• If $k$ is positive: The graph moves upwards.

For example, with the function $f(x) = x^2$, if you want to move it up by 4 units, you write it as:

$$f(x) + 4 = x^2 + 4$$

The shape stays the same, but the vertex now goes from (0,0) to (0,4).

• If $k$ is negative: The graph moves downwards.

Continuing with $f(x) = x^2$, if we shift it down by 5 units, we represent it as:

$$f(x) - 5 = x^2 - 5$$

Now, the vertex changes from (0,0) to (0,-5).

Key Differences

Here’s a quick summary of the main differences between horizontal and vertical translations:

1. Axis of Movement:

   • Horizontal: Moves left or right on the x-axis.
   • Vertical: Moves up or down on the y-axis.

2. Effect on Function:

   • Horizontal translations change the input value $x$.
   • Vertical translations change the output value of the function.

3. Direction of Shift:

   • A positive or negative $h$ shows shifts to the right or left.
   • A positive or negative $k$ shows shifts up or down.

By understanding these translations, you can easily change how graphs look, which is a really helpful skill in Year 12 Mathematics as you start working with more complex problems in graphing and functions.