Function Transformations
One template moves or reshapes any parent function — slide it, flip it, stretch it — without changing its essential personality.
1 The One Template
Every transformation of a parent function f lives in a single template. Learn what each letter does once, and you can transform any function forever.
y = a · f(b(x − h)) + k
h
Shifts left / right
Inside the function → direction is opposite: (x − 3) shifts right 3.
k
Shifts up / down
Outside the function → moves as written: + 2 shifts up 2.
a (if negative)
Stretch / flip vertical
|a| > 1 stretches (narrower); 0 < |a| < 1 compresses (wider); negative flips over the x-axis.
b
Stretch / compress horizontal
Also inside → also opposite-feeling. (Our examples keep b = 1.)
The memory hook: work inside-out, like unwrapping parentheses — the (x − h) shift first, then the multiplier a (reflect/stretch), then the +k shift. Multiplication reshapes; addition relocates.
2 See It
Same DNA (all parabolas), four outfits — and the sneaky inside shift.
Same DNA (all parabolas), four outfits. Sliding never changes the shape; flipping reverses it; the 2 in 2x² squeezes it narrower.
The sneaky one: x + 4 shifts LEFT, not right. Changes inside the function (touching x) move opposite to intuition; changes outside (touching y) move as expected.
3 Worked Examples
Follow each step. The pattern never changes: match the template, read h and k, track the vertex.
Example 1 Shifts: describe y = (x − 3)² + 2 from y = x²
- Match the template: inside you see (x − 3), so h = 3; outside you see + 2, so k = 2.
- Inside shifts go opposite → but (x − 3) already points right: shift right 3. Outside moves as written: shift up 2.
- Track the vertex: (0, 0) → (3, 2).
Shift right 3, shift up 2; vertex (3, 2). Check: x = 3 gives (0)² + 2 = 2. ✓
Example 2 Reflection: describe y = −x² from y = x²
- The negative sign is the “a” multiplier: a = −1.
- Negative a flips the graph over the x-axis; |a| = 1 means no stretch.
- The vertex stays at (0, 0) — flipping doesn’t relocate anything.
Reflect over the x-axis (opens downward instead of up). Check: x = 2 gives −4 instead of 4 — every y-value negated. ✓
Example 3 Stretch vs. compression: y = 2x² and y = (1/2)x²
- |a| > 1 stretches vertically (narrower); 0 < |a| < 1 compresses (wider).
- y = 2x²: vertical stretch by 2 (narrower).
- y = (1/2)x²: vertical compression by 1/2 (wider).
Stretch factor 2 (narrower); compression factor 1/2 (wider). Check: at x = 2, x² = 4, 2x² = 8 (grew), (1/2)x² = 2 (shrank). ✓
Example 4 Combined: describe y = −2(x − 1)² + 3 from y = x²
- Work inside-out: (x − 1) shifts right 1.
- The −2 reflects over the x-axis and stretches vertically by 2.
- The +3 shifts up 3. Vertex: (0, 0) → (1, 3).
Reflect over x-axis, vertical stretch by 2, shift right 1, shift up 3; vertex (1, 3). Check: x = 1 → 3 ✓; x = 2 → −2 + 3 = 1, down from the vertex as expected. ✓
4 Common Mistakes
Three traps that catch nearly everyone. Learn to spot them here instead of on a test.
Mistake 1: Reading inside shifts backwards
Wrong
“y = (x + 4)² shifts right 4” — reading the + as “move right.”Right
(x + 4) = (x − (−4)), so h = −4 → shift LEFT 4.Fix: inside the function, direction is OPPOSITE. Ask “what x-value makes the inside zero?” — x = −4 zeroes (x + 4), so the action happens at −4: the graph moved left.
Mistake 2: Applying transformations in a scrambled order
Wrong
For y = −2(x − 1)² + 3, shifting up 3 before reflecting — then misplacing the vertex.Right
Handle the inside (shift), then a (reflect/stretch), then k (vertical shift) — or just track the vertex through each step: (0,0) → (1,0) → (1,0) reflected/stretched → (1,3).Fix: work inside-out, like unwrapping parentheses: the (x − h) first, then the multiplier a, then the +k.
Mistake 3: Confusing stretch with shift
Wrong
“y = 2x² moves the parabola up 2.”Right
2x² multiplies every y-value by 2 — the vertex stays at (0, 0)! Only +k moves things up/down.Fix: multiplication reshapes (stretch/flip); addition relocates (shift). If the number multiplies the function, the shape changes; if it’s added, the position changes.
5 Quick Checks
Cover the answer, try it, then reveal.
1. Describe y = x² + 3 from y = x². Which way does it shift?
Answer
Shift up 3 (outside change, moves as written).2. Describe y = (x − 5)² from y = x². Direction and distance?
Answer
Shift right 5.3. Starting from y = |x|, is y = −|x| reflected over the x-axis?
Answer
Yes — the V opens downward now.6 Key Points
- y = a · f(b(x − h)) + k — one template for every transformation.
- h shifts left/right (inside → opposite); k shifts up/down (outside → as written).
- a: |a| > 1 vertical stretch, 0 < |a| < 1 compression, negative flips over the x-axis.
- Work inside-out: (x − h) first, then a, then k. Track the vertex through each step.
- Multiplication reshapes (stretch/flip); addition relocates (shift).
- The (x + 4) trap: ask “what x zeroes the inside?” — that is where the graph moved.