Scale factor of sqrt(x)

Question

Hello,

How do I know what the scale factor is for variations on sqrt(x)? Some are less than, some are steeper, but how do i find the number? My homework just shows the regular graph and a second sqrt graph. I know how to make the rest of the equation. Thank you!

Answer 1

The simplest way is to see what the value of function is when the argument of the square root (what's inside the square root) is equal to 1. Let's take a look at an example to see why that is:

y = A·sqrt(x) where A is some constant coefficient (a scale factor).

If x=1, then y=A since sqrt(1)=1. If we look at a graph of this function for a particular value of A, we would see the curve goes through the point (1, A) so whatever the value at x=1 is, is the value of A.

In cases where the argument is more than just x like:

y = A·sqrt(x + 4)

Then when x= -3, sqrt(x + 4)=1 and y=A and we know our curve goes through the point (-3, A).

The last piece that may crop up is if you have some vertical shift:

y = A·sqrt(x + 4) + 1

In this case, we should still go to x=-3 but since everything gets shifted up by 1, we should subtract 1 from the value at x=-3 to get A.

In short to wrap up, go to x coordinate that makes the inside of the square root become 1 and the value of the function there is the scale factor A plus whatever vertical shift there is in the function.