What kind of transformation converts the graph of f(x)=9x^2–9 into the graph of g(x)=x^2–1?

Question

There is only one right answer.

Here are the options...

Vertical stretch

vertical shrink

horizontal stretch

horizontal shrink

Answer 1

First, it's good to solve these mathematically.

We have two functions f(x) and g(x). We can factor out 9 from the right side of f(x), giving us:

f(x) = 9*(x^2-1)

g(x) = (x^2-1)

Or, put another way, f(x) = 9*g(x)

Now we just need to find how to describe this graphically. This is what I would recommend in general: for set value(s) of x, what happens to f, what happens to g, and how are they related.

So, for example, when x is 0, f(0)=-9, g(0) = -1.

Similarly, for x=10, f(10) = 99*9, g(10) = 99

for x=-1, f(-1) = 0, g(-1) = 0

So this confirms for us that f(x) = 9*g(x)

Now let's imagine f(x) and g(x) as graphs, with f and g on the y axis, and x on the x axis.

Because the values of f(x) are "bigger" (by a factor of 9) than the g(x) values, I would call this a vertical stretch when going from g to f, and a vertical shrink going from f to g.

The question is asking about going from f to g, so I'd call it a vertical shrink