Question

If the parent function f(x)= 3sq rootx is transformed to g(x) = 3sq rootx + 2 - 4 , which is the graph of g(x)?

Answer 1

Let's analyze the changes made to the parent function one by one:

STEP 1: Horizontal translation.

If we transform

$\sqrt{x}\mapsto \sqrt{x+2}$

We're performing a change in the form of

$f(x)\mapsto f(x+k)$

This kind of changes result in a horizontal translation, k units to the left if k is positive, k units to the right if k is negative. In this case, k=2, so the original graph is shifted 2 units to the left.

STEP 3: Vertical stretch.

If we transform

$\sqrt{x+2}\mapsto 3\sqrt{x+2}$

We're performing a change in the form of

$f(x)\mapsto kf(x)$

This kind of changes result in a vertical stretch with scale factor k. If k is negative, the function is also reflected across the x axis. In this case, k=3, so the original graph is stretched vertically, with scale factor 3.

STEP 3: Vertical translation.

If we transform

$3\sqrt{x+2}\mapsto 3\sqrt{x+2}-4$

We're performing a change in the form of

$f(x)\mapsto f(x)+k$

This kind of changes result in a vertical translation, k units down if k is positive, k units up if k is negative. In this case, k= -4, so the graph is shifted 4 units down.

All, in all, the original graph is shifted 2 units to the right, then it's stretched vertically with scale 3, and then it's shifted 4 units down. The order is important!