Scaling Functions Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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.

Solution

  1. 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.
  2. 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.
  3. 3
    The shape of the graph (concave down, starting at origin) is preserved; only the vertical scale changes.

Answer

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

About Scaling Functions

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.

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More Scaling Functions Examples

Example 2 medium

Explain the difference between [formula] (horizontal scaling) and [formula] (vertical scaling) for [

Example 3 easy

The graph of [formula] has a maximum at [formula]. Where is the maximum of [formula]? Of [formula]?

Example 4 hard

Starting from [formula], write the equation and describe each transformation for [formula]. State th

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