AS & A-Level Maths: Why Bad Guy x Behaves the Way He Does
Here’s a neat mnemonic for remembering how to transform a graph horizontally for A-level maths: Bad Guy x, whom you usually find in bracket-prison, always does the opposite of what you tell him. For example, if you have a graph like y=f(x–2), Bad Guy x moves the graph of y=f(x) two units to the right, not the left. Again, that’s a neat mnemonic, but it doesn’t do anything to explain why graphs behave that way.
It’s time to put that right and give you an alternative way of thinking about what’s going on. The key thing is to distinguish between the function (a list of instructions that takes an argument and returns a single output) and the graph (a visual representation of the link between x and y values, which in this case involves the function).
So, for example, if you know the ‘old’ graph of y= f(x) and you want to plot the ‘new’ graph of y= f(x–2) as before, you might think, ‘Where does the new graph cross the y-axis?’ Obviously, where x=0. Substituting 0 for x in the graph’s equation gives you y=f(–2) – so the y–value you need is what comes out of the function when the argument is –2. That is to say, the y-value of the new graph when x=0 is the same as the y-value of the old graph when x=–2.
You can use the same trick for any value of x you like: the y-value of the new graph is always the y-value of the old graph two units to the left – which means the new graph is two units to the right of the old graph, the opposite of what you’d expect if you’d never met Bad Guy x.
A similar argument explains why y=f(3x) is a narrower version of y=f(x) rather than a wider one: the y-value of the new graph at any given value corresponds to the y-value of the old graph at an x-value three times as large.
Outside of the bracket, the reasoning is different. If your old graph is y=f(x) and the new graph is y=f(x)+4, then to find the new y-value, you simply add 4 to the old y-value, moving the graph upwards, as you’d expect.
It’s not the x-ness or the y-ness that makes the graphs behave the way they do: it’s the bracketiness. If you compare the general equation of a circle, (x–a)²+ (y–b)²=r², to the equation of a circle centred on the origin, x²+y²=r², you may notice that the circle has been moved right by a and up by b – not down, as you may expect for something related to Good Guy y.