Scaling Functions Math Example 2

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

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.

Solution

  1. 1
    h(x)=2f(x)=2x2h(x)=2f(x)=2x^2: vertical stretch. h(3)=2(9)=18h(3)=2(9)=18. Graph is narrower (taller).
  2. 2
    g(x)=f(2x)=(2x)2=4x2g(x)=f(2x)=(2x)^2=4x^2: horizontal compression by factor 12\frac{1}{2} (argument multiplied by 22). g(3)=(6)2=36g(3)=(6)^2=36.
  3. 3
    Both transform the parabola, but differently: vertical scaling multiplies output; horizontal scaling changes the rate of input. Here g(x)=4x2g(x)=4x^2 while h(x)=2x2h(x)=2x^2, so gg compresses more strongly.

Answer

h(3)=18h(3)=18 (vertical ×2); g(3)=36g(3)=36 (horizontal ÷2, equivalent to ×4 vertically)
Horizontal scaling by bb (replacing xx with bxbx) compresses by 1/b1/b; vertical scaling by cc multiplies outputs by cc. For even functions like x2x^2, horizontal compression by 22 is equivalent to vertical stretch by 44.

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.

Learn more about Scaling Functions →

More Scaling Functions Examples

Example 1 easy

Describe how [formula] and [formula] transform the graph of [formula]. Evaluate both at [formula].

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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