Key Idea: A transformation moves, stretches or flips a graph. A change outside f acts on y and behaves normally. A change inside f acts on x and behaves backwards.
| Change | Effect |
|---|---|
| f(x) + b | Up b |
| p f(x) | p times taller |
| −f(x) | Flip in the x-axis |
| f(x − a) | Right a |
| f(qx) | Width divided by q |
| f(−x) | Flip in the y-axis |
Combined: y = p·f(x − a) + b
- Inside — Do the horizontal changes first.
- Outside — Then the vertical changes: stretch or flip in y, and shift up last.
- Key points — Track a few points, such as maxima and intercepts.
Inside first, then outside, and inside works backwards.
f has a maximum at (4, 5). Where is the maximum of 2f(x − 1) + 3? (5, 13).
Which way does f(x + 2) move the graph? Left 2.
What does f(2x) do to the width? It halves it.
Does f(x − a) change any heights? No. Only the x-values move.