In this series of activities, we will be tying two topics together, transforming a graph that is given both in **function notation [f(x)]** and** cartesian [y = ]**

**How to transform graphs**

When studying shapes, you will have become familiar with the four different types of transformation - reflection, rotation, translation and enlargement.

When you are dealing with transforming graphs, you can **translate**,** reflect **and** stretch**, enlarging in one direction, but not rotation.

In this activity, we are going to look at **reflecting (both horizontally and vertically)**.

**y = f(x) → y = -f(x) and y = f(-x)**

This does like quite daunting at first but it is actually quite intuitive.

f(x) just describes any function - it could be a quadratic, a linear graph, a circle or any other function. For example, here is a random function:

If we apply the transformation** y = -f(x),** every value of **y becomes negative** and the value of x remains the same. This will lead to a** reflection in the x-axis**.

If we apply the transformation **y = f(-x)**, every value of **x becomes negative **and the value of y remains the same. This will lead to a** reflection in the y-axis**.

**How to apply this to a graph**

**Example: **

You are given the function y = x^{2} - 4

**Sketch the graph of y = 4 – x ^{2}**

With this example, you have to rearrange the function y = 4 – x^{2} so that it is a transformation of the original function.

If we factorise out the value of -1, we will get y = 4 – x^{2 }→ y = -( x^{2} - 4)

From this, we can see that** y = 4 – x ^{2 }is a reflection in the x-axis of y = x^{2} - 4**

**Example:**

You are given the function y = x^{2} - 3x - 4

**Sketch the graph of y = x ^{2} + 3x - 4**

This one is a bit more tricky but you still have to rearrange the function y = x^{2} + 3x – 4 so that it is a transformation of the original function y = x^{2} - 3x – 4

This one has to be done by a process of elimination.

You should be able to eliminate all the other transformations, so we try to see if this is y = f(x) → y = f(-x)

y = (- x)^{2} + -3 (- x) – 4

y = x^{2} + 3x – 4

From this, we can see that** y = x ^{2} + 3x – 4 is a reflection in the y-axis of y = x^{2} - 3x – 4**

Time for some questions now.