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Function Transformations Calculator

Online function transformation tool: enter the stretch a and shifts h, k to see how y=a·f(x−h)+k changes the original graph.

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y=a·f(x−h)+k
f(x)f(x−h)
a<0 reflects in x-axis
Right for x−h
Up for +k

📖 Tutorial | Transformation

1. Definition
From y=f(x), shifts and stretches give y=a·f(x−h)+k: a vertical stretch, h horizontal shift, k vertical shift.
2. Formula & Notation
aVertical stretch (a<0 reflects)
hHorizontal shift
kVertical shift
3. Properties & Laws
  • f(x)+k shifts up for k>0;
  • f(x−h) shifts right for h>0;
  • a f(x) stretches vertically by a;
  • f(bx) stretches horizontally by 1/b;
  • −f(x), f(−x) reflect in the axes.
4. Steps
  1. Enter a, h, k;
  2. Click "Calculate" for the description;
  3. Or click "Load Example".
5. Worked Examples
Example: y=2f(x−1)+3: shift right 1, stretch ×2, shift up 3.
6. Common Pitfalls
x−h moves right;
Order of shift and stretch matters;
For negative a, reflect after stretching.

❓ FAQ | Transformation

Basic types?
Shifts, stretches, reflections.
Horizontal?
f(x−h) right for h>0.
Vertical?
f(x)+k up.
a f(x)?
Vertical stretch.
f(bx)?
Horizontal 1/b.
Minus?
Reflect in an axis.
Order?
It matters.