## What you need to know

A **transformation** is something that is done to a graph/function that causes it to change in some way. This topic is about the effects that changing a function has on its graph.

There are two types of transformation you must be familiar with: **translations **and **reflections**. Throughout this topic, we will use the notation f(x) to refer to a function and describe the changes that happen to it.

**Translations**

When a function/graph is translated, it is shifted – left/right, up/down, or some combination of the two. If we let f(x) be any function, then the two types of translation are:

– f(x) + a; in this case, the function adds the value of a on top of all of its outputs. This means that the y values on the graph (the outputs of the function) are all increased by a. As a result, the whole graph is translated by a in the **positive **y **direction** (up).

– f(x + a); in this case, a is added to each input before it is passed into the function. This means that in order to get the same output from f(x + a) as from f(x), the input needed is going to be smaller by a, therefore the x values on the graph (the inputs of the function) are all decreased by a. As a result, the whole graph is translated by a in the **negative **x **direction **(left).

**Example: **Let f(x) = x^2. Sketch, on the same axes, the graphs of f(x), f(x) + 4, and f(x + 1).

Firstly, you should recall the parabolic shape of the graph y=x^2.

Now, in the case of the graph

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

every y coordinate is increased by 4 and thus the whole graph is shifted upwards by 4. In the case of the graph

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

every x coordinate is decreased by 1, and thus the whole graph is shifted left by 1. The resulting graphs should look like those in green and blue (respectively).

**Reflections**

When a function/graph is reflected in a line, everything on one side of the line is flipped over and swapped with everything on the other side of the line. If we let f(x) be any function, then the two types of reflection you’ll see are:

– -f(x); in this case, every positive y value is made negative and every negative y is made positive. As a result, the whole graph is **reflected in the x-axis**.

– f(-x); in this case, every positive x value is made negative and every negative x is made positive. As a result, the whole graph is **reflected in the y-axis**.

**Example: **Let f(x) = x^2 - 3x. Sketch, on the same axes, the graphs of f(x), f(-x), and -f(x).

Firstly, recognise that x^2 - 3x factorises to x(x - 3), therefore f(x) is a positive quadratic with roots at 0 and 3.

Now, in the case of the graph

f(-x)= (-x)^2 - 3(-x) = x^2 +3x,

every positive x coordinate is made negative and vice versa, and thus the whole graph is reflected in the y-axis. In the case of the graph

-f(x) = -(x^2 - 3x) = -x^2 + 3x,

every positive y coordinate is made negative and vice versa, and thus the whole graph is reflected in the x-axis. The resulting graphs should look like those in green and blue (respectively).

In both of these cases, you can verify that the translations are correct by looking at the equations themselves and plotting them like normal quadratics. For example, the first translation, x^2 + 3x, factorises to x(x + 3) which is a positive quadratic with roots at 0 and -3. Indeed, looking at the result of the reflection, we can see that the blue quadratic is precisely a positive quadratic with roots at 0 and -3. Try the second reflection for yourself.

In summary: letting f(x) be any function, the 4 types of transformations you will see are

f(x) + a; translation by +a in the y-direction;

f(x + a); translation by -a in the x-direction;

f(-x); reflection in the y-axis;

-f(x); reflection in the x-axis.

Bear in mind, there is no reason why a question might ask you to combine more than one transformation. In this instance, you should do the reflection first and the translation second.

Additionally, sometimes you will be given a graph of a function f(x) without the equation, only being told some of the coordinates. If you’ve read through this topic, you’ll know that each graph transformation is just a result of coordinates being altered, so in this situation you’ll need to change the coordinates appropriately and let the graph follow.

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### Example Questions

1) Let f(x) = \dfrac{1}{x}. Sketch, on the same axes, the graphs of f(x) and f(x - 3).

If you don’t recall the shape of the reciprocal graph, click here (https://mathsmadeeasy.co.uk/gcse-maths-revision/plotting-quadratic-and-harder-graphs-gcse-revision-and-worksheets/).

Now, f(x - 3) = \dfrac{1}{x - 3} is a translation of positive 3 in the x-direction. So, we get

2) Let f(x) = x^2 - x - 6. Sketch, on the same axes, the graphs of f(x) and -f(x) + 4.

Firstly, x^2 - x - 6 factorises to (x - 3)(x + 2), so it is a positive quadratic with roots at 3 and -2.

Now, -f(x) + 4 is both a reflection in the x-axis and a translation of 4 in the positive y-direction. We should do the reflection first and the translation second – it often helps to sketch the intermediate step to help you, and you can always rub it out afterwards. Here, the dotted line will be the intermediate step (the reflection before the translation). So, we get

Bonus questions: what would happen if you did the translation first and then the reflection? How would this answer be different, or would it be different at all?

Bonus questions: what would happen if you did the translation first and then the reflection? How would this answer be different, or would it be different at all?

3) Below is a graph of f(x) with 3 coordinates A, B, and C marked on. Sketch f(-x), and mark on the new locations of A, B, and C using A’, B’ and C’ respectively.

So, f(-x) means a reflection in the y-axis. In terms of the individual coordinates, the negative x values will be made positive and the positive x values will be made negative. So, we get

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