Amplitude Examples in Math

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

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

Concept Recap

Amplitude is the maximum vertical distance from the midline of a periodic function to a peak or trough.

Amplitude is the maximum displacement from the middle of a wave — it is half the total height of a full oscillation from crest to trough.

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: Amplitude = |a| in f(x) = a\sin(bx + c) + d. It scales the output vertically — it is the vertical scaling factor for the oscillation.

Common stuck point: Amplitude is always non-negative — a negative coefficient like -3\sin(x) gives amplitude 3, not -3; the negative reflects the graph but the amplitude is |-3| = 3.

Sense of Study hint: Find max and min, compute half of their difference.

Worked Examples

Example 1

easy
Find the amplitude of f(x) = 5\sin(x).

Solution

  1. 1
    The general form of a sine function is f(x) = A\sin(Bx + C) + D, where |A| is the amplitude.
  2. 2
    Here A = 5, so the amplitude is |5| = 5.
  3. 3
    This means the graph oscillates between y = -5 and y = 5.

Answer

\text{Amplitude} = 5
Amplitude is the distance from the midline to the maximum (or minimum) of a periodic function. For y = A\sin(x) or y = A\cos(x), the amplitude is |A|. It measures how far the wave deviates from its center position.

Example 2

medium
Find the amplitude and midline of g(x) = -3\cos(2x) + 4.

Practice Problems

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

Example 1

medium
A sinusoidal function has a maximum value of 8 and a minimum value of 2. Find the amplitude and midline.

Example 2

hard
Write the equation of a cosine function with amplitude 4, period \pi, midline y = -1, and a maximum at x = 0.
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Background Knowledge

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

periodic functionstransformationscaling functions