3.5 Transformation of functions  (Page 7/21)

Because the population is always twice as large, the new population’s output values are always twice the original function’s output values. Graphically, this is shown in [link] .

If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.

The following shows where the new points for the new graph will be located.

Symbolically, the relationship is written as

This means that for any input $t,$ the value of the function $Q$ is twice the value of the function $P.$ Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, $t,$ stay the same while the output values are twice as large as before.

The formula $g(x)=\frac{1}{2}f(x)$ tells us that the output values of $g$ are half of the output values of $f$ with the same inputs. For example, we know that $f(4)=3.$ Then

We do the same for the other values to produce [link] .

When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that $g(2)=2.$ With the basic cubic function at the same input, $f(2)=2^3=8.$ Based on that, it appears that the outputs of $g$ are $\frac{1}{4}$ the outputs of the function $f$ because $g(2)=\frac{1}{4}f(2).$ From this we can fairly safely conclude that $g(x)=\frac{1}{4}f(x).$

We can write a formula for $g$ by using the definition of the function $f.$