Scaling Functions Formula

The Formula

y = c \cdot f(x) stretches vertically by factor |c|; reflects over x-axis if c < 0

When to use: Vertical scaling stretches or squishes the graph up/down; horizontal scaling stretches or squishes it left/right. Both change the function's measurements without altering its fundamental character.

Quick Example

f(x) = x^2; \quad 3f(x) = 3x^2 Graph is 3 times as tall at every point.

Notation

c \cdot f(x): |c| > 1 stretches, 0 < |c| < 1 compresses. f(cx): |c| > 1 compresses horizontally, 0 < |c| < 1 stretches.

What This Formula Means

Scaling a function multiplies its output by a constant (vertical scaling) or compresses/stretches its input (horizontal scaling), changing amplitude or period without changing the shape.

Vertical scaling stretches or squishes the graph up/down; horizontal scaling stretches or squishes it left/right. Both change the function's measurements without altering its fundamental character.

Formal View

g(x) = c\,f(x): if f(a) = 0 then g(a) = 0; g(x) = f(cx): g has period \frac{p}{|c|} if f has period p

Worked Examples

Example 1

easy
Describe how g(x)=3f(x) and h(x)=\frac{1}{2}f(x) transform the graph of f(x)=\sqrt{x}. Evaluate both at x=4.

Solution

  1. 1
    g(x)=3\sqrt{x}: vertical stretch by factor 3. All y-values triple. g(4)=3\cdot2=6.
  2. 2
    h(x)=\frac{1}{2}\sqrt{x}: vertical compression by factor \frac{1}{2}. All y-values halve. h(4)=\frac{1}{2}\cdot2=1.
  3. 3
    The shape of the graph (concave down, starting at origin) is preserved; only the vertical scale changes.

Answer

g(4)=6 (stretched); h(4)=1 (compressed)
Multiplying a function by a constant c scales it vertically: if |c|>1, the graph stretches away from the x-axis; if 0<|c|<1, it compresses toward the x-axis. The x-intercepts remain unchanged.

Example 2

medium
Explain the difference between g(x)=f(2x) (horizontal scaling) and h(x)=2f(x) (vertical scaling) for f(x)=x^2. Compare at x=3.

Common Mistakes

  • Thinking 2f(x) shifts the graph up by 2 — it multiplies all outputs by 2 (vertical stretch), not addition
  • Confusing f(2x) with 2f(x) — f(2x) compresses horizontally; 2f(x) stretches vertically; they are completely different
  • Forgetting that scaling preserves zeros — if f(a) = 0, then cf(a) = 0 for any constant c; x-intercepts don't move under vertical scaling

Why This Formula Matters

Scaling transformations are the key to adjusting amplitude and period in wave models — critical for fitting sinusoidal functions to physical measurements.

Frequently Asked Questions

What is the Scaling Functions formula?

Scaling a function multiplies its output by a constant (vertical scaling) or compresses/stretches its input (horizontal scaling), changing amplitude or period without changing the shape.

How do you use the Scaling Functions formula?

Vertical scaling stretches or squishes the graph up/down; horizontal scaling stretches or squishes it left/right. Both change the function's measurements without altering its fundamental character.

What do the symbols mean in the Scaling Functions formula?

c \cdot f(x): |c| > 1 stretches, 0 < |c| < 1 compresses. f(cx): |c| > 1 compresses horizontally, 0 < |c| < 1 stretches.

Why is the Scaling Functions formula important in Math?

Scaling transformations are the key to adjusting amplitude and period in wave models — critical for fitting sinusoidal functions to physical measurements.

What do students get wrong about Scaling Functions?

Factor outside affects y; factor inside affects x (oppositely).

What should I learn before the Scaling Functions formula?

Before studying the Scaling Functions formula, you should understand: transformation.

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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