Section 3.5

14. Scale changes of parabolas can be considered as horizontal scale changes, or vertical scale changes, or both. Refer to the graphs in the book on page 192 (or graph f and g yourself on your graphing calculator).

14a. Under what Horizontal scale change is g the image of f?

Answer: The question can be restated as, What must a be such that

[f under a horizontal scale change of a]

is the same relation as [g]

By inspection, it is clear that 3 is the only possibility for a.

so the answer is

14b. Under what vertical scale change is g the image of f?

Answer: This time, the question can be restated as, What must b be such that

[f under a vertical scale change of b]

is the same relation as [g]

This time, we can not simply READ OFF the answer for b, without doing much rewriting to g to make it resemble our transformed f.

.

And so b must be one-ninth. Making the answer:

(either of those LAST TWO expressions in the line above would work as a final answer).

14c. Under what size change is g the image of f ?

by far the hardest of the three.

Because we are dealing with both a vertical and a horizontal scale change we COULD restate the question as:

What must a and b be such that

is the same relation as

However stating it this way we haven’t USED THE FACT that we’re looking for a SIZE CHANGE, which means that a=b. So, restate the question instead as

What must a be such that

is the same relation as

It would be tricky to figure out how to rewrite the expression on the right so that it’s in the form on the left. So instead we’ll start with , and try to package it up so that it’s in the same form as .

and so,

Giving the answer of