Transformations: shift, stretch, reflect
What you'll learn
Predict how a·f(x − h) + k moves and reshapes any graph — without re-plotting a single point.
Here's the payoff that saves hours: once you know the shape of a basic function — a "parent" like y = x² — you can draw a whole family of related graphs without plotting a single new point. You just move the parent.
The four moves
Every transformation lives in one general form:
y = a · f(x − h) + k
Each letter does one job, and only one:
| Symbol | Move | Direction |
|---|---|---|
| k | vertical shift | up if k > 0, down if k < 0 |
| h | horizontal shift | right if h > 0, left if h < 0 |
| a | vertical stretch | taller if |a| > 1, flatter if |a| < 1 |
| −a | reflection | flips over the x-axis |
The one that trips everyone up
Vertical shifts feel natural: + k moves the graph up. But horizontal shifts run backwards from what the sign suggests:
- y = (x − 2)² shifts right by 2.
- y = (x + 2)² shifts left by 2.
Why? Ask what input makes the inside zero — the vertex of the parent. For (x − 2)², that's x = 2, so the vertex slid to the right. The minus sign inside subtracts before f acts, so the graph has to wait longer (larger x) to reach the same shape.
Reading one off
For y = −2(x + 3)² + 1, read it piece by piece:
- − → flipped upside down
- 2 → twice as tall (steeper)
- (x + 3) → shifted left 3
- + 1 → shifted up 1
So the vertex is at (−3, 1), opening downward. No plotting required.
Why this matters
Transformations turn a handful of parent shapes — lines, parabolas, √x, 1/x, |x|, sine — into every graph you'll meet. Spot the parent, read off a, h, and k, and you've drawn the graph in your head.
Check your understanding
Question 1 of 2
How does the graph of y = (x − 4)² compare to y = x²?