Determine amplitude of a sinusoidal function

Amplitude is the positive distance from a sinusoidal function’s midline to either maximum or minimum, equivalently half the difference between its maximum and minimum values. For $y=a\sin(b(x-h))+d$ or $y=a\cos(b(x-h))+d$ , the amplitude is $|a|$ , while $b$ , $h$ , and $d$ affect other features; a negative $a$ reflects the graph but does not make amplitude negative. This scope concerns constant-amplitude sine and cosine functions, not damping or other varying-amplitude models.

Detailed Explanation: Determine amplitude of a sinusoidal function

Amplitude measures how far a sinusoidal graph moves above or below its midline. It is always positive.

For a function written as

y=a\sin(b(x-h))+d

y=a\cos(b(x-h))+d,

the amplitude is

\boxed{ \vert a \vert }.

The number $b$ affects the period, $h$ affects the horizontal shift, and $d$ affects the midline. A negative value of $a$ reflects the graph, but the amplitude is still positive.

Example

Determine the amplitude of

y=-3\cos\left(2(x-1)\right)+5.

Step 1: Identify $a$ .

Compare the function with

y=a\cos(b(x-h))+d.

The coefficient of the cosine is

a=-3.

Step 2: Take the absolute value.

\text{Amplitude}= \vert a \vert = \vert -3 \vert =3.

Therefore, the amplitude is

\boxed{3}.

The graph’s midline is $y=5$ , so it reaches $3$ units above and below that line. Its maximum is $8$ and its minimum is $2$ , which also confirms the result:

\frac{8-2}{2}=3.

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Sinusoidal Function Parameters (1 Param) - Function to Parameters

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