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 is the distance from the midline of a wave up to a peak (or down to a trough).

Common stuck point: The procedure for amplitude is the easy part; the trap is using the full crest-to-trough height as amplitude. Asking "Am I measuring the vertical distance from the midline to a peak (half the total swing)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring the vertical distance from the midline to a peak (half the total swing)?

Worked Examples

Example 1

easy

Find the amplitude of

f(x) = 5\sin(x)

Answer

\text{Amplitude} = 5

First step

The general form of a sine function is

f(x) = A\sin(Bx + C) + D

, where

|A|

is the amplitude.

Full solution

2
Here $A = 5$ , so the amplitude is $|5| = 5$ .
3
This means the graph oscillates between $y = -5$ and $y = 5$ .

Amplitude is the distance from the midline to the maximum (or minimum) of a periodic function. For

y = A\sin(x)

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

Example 3

medium

Find the amplitude, period, and midline of

y = 2\sin(3x) - 4

Example 4

medium

A Ferris wheel rises from

2

m to

42

m. Find the amplitude and midline of the height function.

Example 5

hard

Express

y = 3\sin(x) + 4\cos(x)

R\sin(x + \phi)

and identify the amplitude.

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

Example 3

easy

Find the amplitude of

y = 4\sin(x)

Example 4

easy

Find the amplitude of

y = 7\cos(x)

Example 5

easy

Find the amplitude of

y = -3\sin(x)

Example 6

easy

What is the amplitude of

y = \sin(x)

Example 7

easy

Find the amplitude of

y = \frac{1}{2}\cos(x)

Example 8

easy

Find the amplitude of

y = 5\sin(3x)

Example 9

easy

Find the amplitude of

y = 2\sin(x) + 5

Example 10

easy

Find the amplitude of

y = -\cos(x)

Example 11

medium

A sinusoid has maximum value

9

and minimum value

1

. Find its amplitude.

Example 12

medium

A sinusoid has maximum

12

and minimum

-4

. Find its amplitude.

Example 13

medium

Write a sine function with amplitude

6

and midline

y = 0

Example 14

medium

Find the amplitude of

y = -\frac{3}{4}\sin(2x) - 1

Example 15

medium

A wave oscillates between

y = 3

and

y = 7

. What is its amplitude?

Example 16

medium

The amplitude of

y = A\cos(x)

5

and

A < 0

. Find

A

Example 17

medium

Find the amplitude of

y = 3\sin(x) - 3\cos(x)

Example 18

medium

A function has midline

y = 2

and reaches a maximum of

11

. Find its amplitude.

Example 19

medium

Find the amplitude of

y = 10 - 4\cos(x)

Example 20

challenge

A Ferris wheel's height is

h = 25 - 20\cos(t)

meters. How far above its lowest point is the highest point?

Example 21

challenge

A sinusoid passes through a maximum of

8

at one point and a minimum of

-2

. Write its amplitude and midline.

Example 22

challenge

Two waves

y_1 = 3\sin(x)

and

y_2 = 4\cos(x)

are added. Find the amplitude of

y_1 + y_2

Example 23

easy

Find the amplitude of

y = 8\sin(x)

Example 24

easy

Find the amplitude of

y = -6\cos(x)

Example 25

easy

Find the amplitude of

y = \frac{2}{3}\sin(4x)

Example 26

easy

What is the amplitude of

y = \cos(x) + 7

Example 27

easy

Find the amplitude of

y = -\frac{5}{2}\cos(x) + 3

Example 28

medium

A sinusoid has midline

y = -2

and amplitude

7

. What are its maximum and minimum values?

Example 29

medium

Write a cosine function with amplitude

3

, midline

y = 1

, and period

2\pi

that has a maximum at

x = 0

Example 30

medium

Find the amplitude of

y = 5\sin(x) + 12\cos(x)

Example 31

medium

Find the amplitude of

y = -8\cos(2x) + 1

Example 32

medium

Find the amplitude of

y = \sqrt{3}\sin(x) + \cos(x)

Example 33

medium

y = A\sin(x)

has

A > 0

and reaches a max of

11

, find

A

Example 34

medium

Find the amplitude of

y = 6\sin\!\left(\tfrac{x}{4}\right) - 2

Example 35

hard

A Ferris wheel of radius

15

m has its center

18

m above ground. Write the height function

h(t)

if the wheel starts at the bottom and has period

40

Example 36

hard

A sinusoid passes through

(0, 5)

and

(\pi, -1)

with these as max and min. Find amplitude and midline.

Example 37

hard

Find the amplitude of

y = 7\sin(x) - 24\cos(x)

Example 38

hard

A spring's displacement is

x(t) = 0.4\cos(5t)

. What is its amplitude in cm if

x

is in m?

Example 39

challenge

A sinusoid has amplitude

A

, midline

M

. It passes through points

(0, 7)

at maximum and

(2, 1)

at minimum. Find

A

and

M

Example 40

challenge

Two sinusoids of the same period have amplitudes

5

and

12

and are

90°

out of phase. What is the amplitude of their sum?

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Related Concepts

Periodic Functions Function Transformation Scaling Functions

Background Knowledge

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

periodic functionstransformationscaling functions