Scaling Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scaling Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: a \cdot f(x) scales outputs by a (vertical); f(bx) scales inputs inversely by b (horizontal). Note: f(bx) with b > 1 compresses the graph horizontally.

Common stuck point: Factor outside affects y; factor inside affects x (oppositely).

Sense of Study hint: Compare tables of f(x) and c*f(x) side by side. Notice the x-intercepts stay the same but all other y-values are multiplied by c.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
The graph of f has a maximum at (2, 5). Where is the maximum of y=4f(x)? Of y=f(3x)?

Example 2

hard
Starting from f(x)=\sin(x), write the equation and describe each transformation for g(x)=3\sin(2x). State the amplitude and period of g.
← Back to Scaling Functions Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

transformation