Amplitude

Functions
definition

Also known as: wave height

Grade 9-12

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Amplitude is the maximum vertical distance from the midline of a periodic function to a peak or trough. Amplitude determines how large a wave is — in physics it corresponds to energy (sound loudness, wave intensity), and in modeling it sets the scale of oscillation.

Definition

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

💡 Intuition

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.

🎯 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.

Example

f(x) = 3\sin(x) has amplitude 3: it oscillates between -3 and +3. Doubling amplitude doubles the peak height but does not change the period.

Formula

A= rac{y_{max}-y_{min}}{2}

Notation

In y=Asin(Bx+C)+D, amplitude is |A|.

🌟 Why It Matters

Amplitude determines how large a wave is — in physics it corresponds to energy (sound loudness, wave intensity), and in modeling it sets the scale of oscillation.

💭 Hint When Stuck

Find the maximum and minimum y-values of the function. Amplitude equals half the difference: A = (y_{\max} - y_{\min}) / 2. For y = a\sin(bx) + d, the amplitude is simply |a|.

Formal View

Amplitude can be formalized with precise domain conditions and rule-based inference.

🚧 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.

⚠️ Common Mistakes

  • Reading amplitude as the coefficient including its sign — amplitude is always positive: |-3| = 3, not -3
  • Computing the full peak-to-trough distance instead of half — amplitude is HALF the total vertical span, not the full distance from max to min
  • Confusing amplitude with vertical shift — in y = 3\sin(x) + 5, the amplitude is 3 (the stretch) and the vertical shift is 5 (the midline)

Frequently Asked Questions

What is Amplitude in Math?

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

What is the Amplitude formula?

A= rac{y_{max}-y_{min}}{2}

When do you use Amplitude?

Find the maximum and minimum y-values of the function. Amplitude equals half the difference: A = (y_{\max} - y_{\min}) / 2. For y = a\sin(bx) + d, the amplitude is simply |a|.

How Amplitude Connects to Other Ideas

To understand amplitude, you should first be comfortable with periodic functions, transformation and scaling functions.

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