Why does adding a constant to a function's input (as in f(x-a)) shift the plot of the function along the x-axis?

Imagine the function f(x) as a black box which takes in any value x and produces an output y. The black box acts on its input according to a rule which produces a unique value of y for a given x. If the black box becomes f(x-a) then whenever we throw a value into it a is subtracted from this value before the usual rule is applied. Thus if we imagine a plot of the function we see that any given point on the x-axis will become associated with the y-value originally paired with an adjacent point on the x-axis separated by a distance a.

For example, if we throw 4 into f(x-2)=(x-2)^2 we get 4. But we also get 4 if we throw 2 into f(x)=x^2. Similarly, we get the same output from inputting 2 into f(x-2) as from inputting 0 into f(x) and so on. Drawing an example like f(x)=x^2 and trying some example inputs and outputs should allow you to visualise how this will shift the plot depending on the value of the constant a.

Answered by James H. Maths tutor

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