Example 15 from Algebra and Trigonometry, 3.5 Transformation of Functions

Recognizing a Vertical Stretch

The graph in Figure 22 is a transformation of the toolkit function $f (x) = x^{3} .$ Relate this new function $g (x)$ to $f (x),$ and then find a formula for $g (x) .$

Figure 22

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
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 $f .$

$g (x) = \frac{1}{4} f (x) = \frac{1}{4} x^{3}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.

How did it go?

Solution

Your turn

Solution

Your turn