Function Transformation Formula

The Formula

y = a \cdot f(b(x - h)) + k where a = vertical stretch, b = horizontal compression, h = horizontal shift, k = vertical shift

When to use: Moving or reshaping a graph without changing its basic shape.

Quick Example

f(x) + 2 shifts up 2.
f(x - 3) shifts right 3.
2f(x) stretches vertically.
-f(x) reflects.

Notation

Parent function f(x) is transformed: +k shifts up, -h shifts right, a scales vertically, b scales horizontally.

What This Formula Means

A function transformation shifts, stretches, compresses, or reflects the graph of a parent function by modifying its formula in a systematic way.

Moving or reshaping a graph without changing its basic shape.

Formal View

g(x) = a\,f(b(x - h)) + k: vertical scale |a|, reflect if a < 0; horizontal scale \frac{1}{|b|}, reflect if b < 0; shift right h, up k

Worked Examples

Example 1

easy
Describe all transformations applied to f(x) = x^2 to obtain g(x) = 2(x-3)^2 + 1.

Solution

  1. 1
    Write in standard form y = a\cdot f(b(x-h))+k: here a=2, b=1, h=3, k=1.
  2. 2
    Horizontal shift: h=3 shifts the parabola 3 units to the right (vertex moves from (0,0) to (3,0)).
  3. 3
    Vertical stretch: a=2 stretches vertically by factor 2 (makes parabola narrower). Vertical shift: k=1 shifts the entire graph 1 unit up. Final vertex: (3, 1).

Answer

Shift right 3, stretch vertically by 2, shift up 1; vertex at (3,1)
The transformation y = a\cdot f(b(x-h))+k encodes four independent transformations. Reading off a, b, h, k allows systematic description without re-deriving the graph from scratch.

Example 2

medium
Starting from f(x) = \sqrt{x}, apply the transformation g(x) = -\sqrt{2x+4} step by step and identify the key point transformations.

Common Mistakes

  • Thinking f(x - 3) shifts left — it actually shifts RIGHT 3 units; horizontal transformations act opposite to the sign
  • Confusing vertical and horizontal stretches — 2f(x) stretches vertically; f(2x) compresses horizontally
  • Applying multiple transformations in the wrong order — horizontal transformations (inside) apply before vertical (outside) in most standard forms

Why This Formula Matters

Understand any function as a transformation of a basic function.

Frequently Asked Questions

What is the Function Transformation formula?

A function transformation shifts, stretches, compresses, or reflects the graph of a parent function by modifying its formula in a systematic way.

How do you use the Function Transformation formula?

Moving or reshaping a graph without changing its basic shape.

What do the symbols mean in the Function Transformation formula?

Parent function f(x) is transformed: +k shifts up, -h shifts right, a scales vertically, b scales horizontally.

Why is the Function Transformation formula important in Math?

Understand any function as a transformation of a basic function.

What do students get wrong about Function Transformation?

f(x - 3) shifts RIGHT (opposite of the sign). f(x) - 3 shifts DOWN.

What should I learn before the Function Transformation formula?

Before studying the Function Transformation formula, you should understand: function definition, coordinate plane.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus →
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