Example 13 from Algebra and Trigonometry, 3.5 Transformation of Functions

Graphing a Vertical Stretch

A function $P (t)$ models the population of fruit flies. The graph is shown in Figure 20.

Graph to represent the growth of the population of fruit flies.

Figure 20

A scientist is comparing this population to another population, $Q,$ whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
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 Figure 21.

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.

$\begin{array}{l} (0, 1) \to (0, 2) \\ (3, 3) \to (3, 6) \\ (6, 2) \to (6, 4) \\ (7, 0) \to (7, 0) \end{array}$

Figure 21

Symbolically, the relationship is written as

$Q (t) = 2 P (t)$
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.

How did it go?

Solution

Solution