Scaling Functions Formula

The scaling functions formula uses multipliers to stretch or compress a graph: a x f(x) stretches vertically by factor |a| (and reflects if a < 0).

The Formula

y=c⋅f(x)y = c \cdot f(x) stretches vertically by factor ∣c∣|c|; reflects over xx-axis if c<0c < 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)=x2;3f(x)=3x2f(x) = x^2; \quad 3f(x) = 3x^2 Graph is 3 times as tall at every point.

Notation

c⋅f(x)c \cdot f(x): ∣c∣>1|c| > 1 stretches, 0<∣c∣<10 < |c| < 1 compresses. f(cx)f(cx): ∣c∣>1|c| > 1 compresses horizontally, 0<∣c∣<10 < |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)g(x) = c\,f(x): if f(a)=0f(a) = 0 then g(a)=0g(a) = 0; g(x)=f(cx)g(x) = f(cx): gg has period p∣c∣\frac{p}{|c|} if ff has period pp

Worked Examples

Example 1

easy
Describe how g(x)=3f(x)g(x)=3f(x) and h(x)=12f(x)h(x)=\frac{1}{2}f(x) transform the graph of f(x)=xf(x)=\sqrt{x}. Evaluate both at x=4x=4.

Answer

g(4)=6g(4)=6 (stretched); h(4)=1h(4)=1 (compressed)

First step

1
g(x)=3xg(x)=3\sqrt{x}: vertical stretch by factor 33. All yy-values triple. g(4)=3⋅2=6g(4)=3\cdot2=6.

Full solution

  1. 2
    h(x)=12xh(x)=\frac{1}{2}\sqrt{x}: vertical compression by factor 12\frac{1}{2}. All yy-values halve. h(4)=12⋅2=1h(4)=\frac{1}{2}\cdot2=1.
  2. 3
    The shape of the graph (concave down, starting at origin) is preserved; only the vertical scale changes.
Multiplying a function by a constant cc scales it vertically: if ∣c∣>1|c|>1, the graph stretches away from the xx-axis; if 0<∣c∣<10<|c|<1, it compresses toward the xx-axis. The xx-intercepts remain unchanged.

Example 2

medium
Explain the difference between g(x)=f(2x)g(x)=f(2x) (horizontal scaling) and h(x)=2f(x)h(x)=2f(x) (vertical scaling) for f(x)=x2f(x)=x^2. Compare at x=3x=3.

Example 3

medium
Compare g(x)=f(3x)g(x) = f(3x) and h(x)=3f(x)h(x) = 3 f(x) when f(x)=xf(x) = \sqrt{x}. Evaluate both at x=9x = 9.

Common Mistakes

  • Applying the horizontal rule like the vertical one - f(2x)f(2x) compresses (factor 12\tfrac12), it does not stretch.
  • Confusing amplitude (vertical) with period (horizontal) - c f(x)c\,f(x) changes amplitude; f(cx)f(cx) changes period.
  • Forgetting a negative multiplier also reflects - ∣c∣|c| scales while the sign of cc flips the graph.

Why This Formula Matters

Scaling is half of the transformation toolkit (the other half is shifting), essential for graphing sinusoids' amplitude and period and any parent-function variant. Mixing up inside vs. outside, or stretch vs. compress, garbles every transformed graph a student draws. Recognizing it by "Is the function the same shape multiplied by a constant on the output or input (not shifted)?" — rather than by familiar numbers — is what lets a student tell it apart from shifting functions and vertical vs. horizontal scaling and reflecting functions in a mixed problem set.

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⋅f(x)c \cdot f(x): ∣c∣>1|c| > 1 stretches, 0<∣c∣<10 < |c| < 1 compresses. f(cx)f(cx): ∣c∣>1|c| > 1 compresses horizontally, 0<∣c∣<10 < |c| < 1 stretches.

Why is the Scaling Functions formula important in Math?

Scaling is half of the transformation toolkit (the other half is shifting), essential for graphing sinusoids' amplitude and period and any parent-function variant. Mixing up inside vs. outside, or stretch vs. compress, garbles every transformed graph a student draws. Recognizing it by "Is the function the same shape multiplied by a constant on the output or input (not shifted)?" — rather than by familiar numbers — is what lets a student tell it apart from shifting functions and vertical vs. horizontal scaling and reflecting functions in a mixed problem set.

What do students get wrong about Scaling Functions?

The procedure for scaling functions is the easy part; the trap is applying the horizontal rule like the vertical one. Asking "Is the function the same shape multiplied by a constant on the output or input (not shifted)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 →
← Back to Scaling Functions See All Examples Practice Problems