
Transformations
How would you solve this question:
The graph of y=f(x) is expanded vertically about the line y=k by a factor of 5. The point (3,2) on the graph of y=f(x) has its image under the transformation at the point (3,22). Find the value of k.
The only way I know how to solve this is with guess and check. >_<
Is there a faster and easier way to solve it?

Well I don't know about faster, but definitely more elegant. This expansion will cause all points to be five times further away from y=k. So, a point was d away from k, then after expansion it will be 5d away from k. So let's take the point they give us, (3,2) goes to (3,22).
Now since 2>22 (ie. it increases), then 2>k. Otherwise 2>a smaller number. So the distance from (3,2) to y=k is 2k. Likewise, the distance from (3,22) to y=k is 22k. This must be 5 times 2k so:
22k=5(2k)
22k=105k
4k=12
k=3