Transforming graphs

This chapter explores graphs transformation. It covers effects which occur when the graphs of f(x) is transformed such as y=af(x), y=f(x)+a, y=f(x+a) and y=f(ax). You must have prior knowledge of basic shapes of graphics for such as quadratics, cubic, and reciprocals.
Transforming a graph is changing aspects of it such as moving it along its axis or by scaling it using a scale factor enlargement. The topic of graph transformation is easy. You must know that;

…also

By writing f(x) in maths allows us to keep track of which value we use for x. This is shown in the example below;




This notation is used throughout this topic. In the examples below a different graph will be used for each transformation but the transformations apply to all different types of graphs.

f(x) + a

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The above is the graph f(x) = x². In this example we’re going to explore graphs in the template of f(x)+a. The graph function above f(x) = x² can also be written as f(x) = x² + 0. Let a = 0. Suppose we change the value of a to observe what happens to the graph.
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Above we realise that by changing the value of a moves the graph up and down by a units as shown in the following animation;
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af(x)

Now let us explore graphs of af(x). Here we shall explore what effects does changing a have on af(x) graph. Below is the graph f(x)=x²
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Let us change the value of 3 or a to observe what happens to the graph.
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Above we realise that by changing the value of a stretches the graph in the y-axis by a scale factor. This graph transformation is shown below;
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f(x+a)

Now lets observe what happen when the value of a is changed in this function. Below is the graph of f(x+5) = (x+5)³. Let us change the value of 5 over and over to observe what happens.
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Above we can see that changing the value of a in f(x+a) moves the graph from left to right by a units as shown in the animation below;
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f(ax)

Now let us observe what effect takes when changing a in f(ax). Below is the graph of f(x) = x² or f(1x)=(1x)². Let 1 be a, we’re going to keep changing this value to observe what happens. We shall begin with a graph of f(5x) = (5x)².
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Summary

In the examples above we’ve seen how values in a function affect a graph by transforming it. Below are some of these transformations we’ve seen above.
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Changing this value transforms the graph by moving it up and down.
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Changing this value transforms the graph by stretching it in the y-axis by a scale factor of a.
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Changing this value transforms the graph by moving it left to right by a units.
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Changing this value stretches the graph in the x-axis by a scale factor of a.