Inside vs. outside (transformation rule)
SAT glossary · Math
Definition
Outside the parentheses does what it says (vertical, forwards): f(x)+c shifts up, -f(x) reflects over the x-axis, c · f(x) stretches. Inside does the opposite (horizontal, backwards): f(x+c) shifts LEFT, f(cx) with c>1 compresses. Do not memorize — ask where the vertex went: the vertex of f(x+3)=(x+3)^2 is where x+3=0, i.e. x=-3.
Inside vs. outside (transformation rule) in plain English
When you transform a graph by changing what's INSIDE the function's parentheses versus what's OUTSIDE, the two behave in opposite, almost backwards-feeling ways. Stuff OUTSIDE the parentheses does exactly what it looks like it should (vertical moves, in the 'normal' direction): f(x)+c shifts the whole graph UP, -f(x) flips it upside down, and c· f(x) stretches it taller. But stuff INSIDE the parentheses does the OPPOSITE of what it looks like (horizontal moves, backwards): f(x+c) actually shifts the graph LEFT (not right, even though you're adding), and f(cx) with c bigger than 1 actually squeezes the graph narrower. Instead of memorizing rules, the safest method is tracking where the vertex (or a key point) physically lands — for f(x+3)=(x+3)^2, the vertex is wherever x+3=0, which is x=-3, confirming the shift really did go left.
Also called: transformation rule
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