Question

Can someone show me how to work out this problem?
Suppose that the graph of f contains the point (5,2). Find a point that must be on the graph of g(x)=3f(x+7)

Answer 1

f(x) leads to a value of 2 for an input of x, right? Since you know (5,2) is on f... ?

So f(5) = 2.

g(x) is a transformation of f(x). There are two parts of the transformation. One part shifts the graph left or right, and the other part stretches it.

First, look ONLY at the part inside the parenthesis, x+7, which says that if you put x = 5 inside f(x), it is sort of like putting x = -2 inside the parenthesis... 'cause -2 + 7 = 5.
Does that make sense?

And then the 3 out in front of the f(x+7)... 3f(x+7) says that g(x) will be 3 times bigger than whatever f(x) would have been. So if all you had was something like h(x) = 3f(x), h(x) would just look like f(x) but 3 times taller.

So g(x) = 3f(x+7) is three times taller than f(x), but also shifted to the left by 7 places.

Answer 2

If (5,2) is on f(x), then an input in the parenthesis in g(x) would lead to a value three times larger... so 3 * 2 = 6.

However, this value of 6 doesn't happen at x=5... it happens shifted over to the left by 7 places... so for x=-2, g(x) = 6.

You can see this more directly by putting -2 in as the x value in g(x).
g(-2) = 3f(-2+7) = 3f(5) = 3(2) = 6

But that's why that works... it's stretched 3 times bigger and shifted left by 7 places.

Answer 3

(sorry for the lengthy explanation :) )

Answer 4

oh no worries! Thank you so much for helping me out. I will try to work out the next problem on my homework as this problem as an example.

Answer 5

For example...