Function Transformations: Shift, Stretch, Reflect
The four basic transformations of a function — how to spot them from an equation and predict the new graph in seconds.
The transformation cheat sheet
Given the parent y = f(x), four operations produce all standard transformations:
| Operation | Effect on graph |
|---|---|
f(x) + k | Shift UP by k (down if k < 0) |
f(x − h) | Shift RIGHT by h (sign trap!) |
a · f(x) | Vertical stretch by |a|; reflect over x-axis if a < 0 |
f(b · x) | Horizontal stretch by 1/|b|; reflect over y-axis if b < 0 |
The sign-flip trap inside parentheses
Students see f(x + 3) and assume "shift right." Wrong — that's shift LEFT by 3. Inside the function, the sign reverses your intuition.
Combining transformations
Apply in order: horizontal stretch → horizontal shift → vertical stretch → vertical shift. Example: g(x) = 3·f(2x − 4) + 1 = horizontal compress by 1/2, shift right 2 (NOT 4), vertical stretch by 3, shift up 1.
Check yourself
📌 Identify
If f(x) = x², which is the graph of g(x) = (x + 2)² − 5?
(x + 2)² = f(x − (−2)) means h = −2, shift LEFT 2. The −5 outside shifts DOWN 5.
Practice with CBE-style practice questions
Drill transformations in Pre-Calc Sem A practice.