Transformations of a function, step by step

Change a function's rule and its graph moves.

The question

f(x) = x³, g(x) = (x + 2)³. What happens to the graph?

What happens to the graph?

g(x) = (x + 2)³—inside

bracket = 0atx = −2

Left2

The change is INSIDE the function — it happens to x before f sees it. That changes the INPUT, so the graph moves sideways, not up or down.
To get f's starting value, the bracket has to come out 0, and x + 2 is 0 when x = −2. So g does at x = −2 what f did at x = 0 — 2 steps EARLIER.
Later along the x-axis is to the right and earlier is to the left, so the graph moves left by 2. Note the sign in the bracket is the OPPOSITE of the direction — that is the part worth remembering.

How it works.

01
The change is INSIDE the function — it happens to x before f sees it. That changes the INPUT, so the graph moves sideways, not up or down.
02
To get f's starting value, the bracket has to come out 0, and x + 2 is 0 when x = −2. So g does at x = −2 what f did at x = 0 — 2 steps EARLIER.
03
Later along the x-axis is to the right and earlier is to the left, so the graph moves left by 2. Note the sign in the bracket is the OPPOSITE of the direction — that is the part worth remembering.