Functions of a certain form can be scaled and translated by modifying the constants contained in the expression. Suppose we have some function of x that has not been shifted or scaled in any way.
$$ y = f(x) $$
We can modify that function to be shifted by (h, k) and scaled by the factor a.
$$y = a*f(x-h)+k$$
- Scaling occurs by simply multiplying the expression by some scale factor. In my examples here, that scale factor is a.
- Translation left and right can be accomplished by replacing x with (x-h), where h is how far the function is shifted to the right.
- Translation up and down is accomplished by simply adding a constant k, where k is how far the function is shifted up.
Here is an interactive example showing how a parabolic function may be shifted and scaled. (If the embedded applet isn’t working, you can access it here.)
And here is the scaling of an exponential function.
(If the embedded applet isn’t working, you can access it here.)
You can also play with examples of function shifting and scaling with the Desmos online graphing calculator.