Trigonometric Function Graphs Examples in Math

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

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

Concept Recap

The graphs of $\sin x$ , $\cos x$ , and $\tan x$ as functions of a real variable, characterized by amplitude, period, phase shift, and vertical shift.

If you track the $y$ -coordinate of a point moving around the unit circle and plot it against the angle, you get the sine wave. It's the shape of ocean waves, sound waves, and alternating current. The general form $y = a\sin(bx - c) + d$ lets you control four properties: how tall the wave is ( $a$ , amplitude), how fast it repeats ( $b$ , affecting period), where it starts ( $c$ , phase shift), and its vertical center ( $d$ , vertical shift).

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: Reading off amplitude, period, phase shift, and vertical shift from $y=a\sin(bx-c)+d$ .

Common stuck point: The procedure for trigonometric function graphs is the easy part; the trap is using $b$ directly as the period. Asking "Is the curve a repeating wave whose height, repeat length, and center I can name from $a$ , $b$ , $c$ , $d$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the curve a repeating wave whose height, repeat length, and center I can name from $a$ , $b$ , $c$ , $d$ ?

Worked Examples

Example 1

easy

Identify the amplitude, period, phase shift, and vertical shift of

y=3\sin(2x-\pi)+1

. Write in standard form

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

Answer

Amplitude

=3

, Period

=\pi

, Phase shift

=\pi/2

right, Vertical shift

=1

; range

[-2,4]

First step

Rewrite:

y=3\sin(2x-\pi)+1=3\sin\!\left(2\left(x-\frac{\pi}{2}\right)\right)+1

Full solution

2
Read off parameters: $a=3$ (amplitude), $b=2$ (period $=2\pi/2=\pi$ ), $h=\pi/2$ (phase shift right $\pi/2$ ), $k=1$ (vertical shift up $1$ ).
3
Summary: oscillates between $k-|a|=1-3=-2$ and $k+|a|=1+3=4$ , with period $\pi$ , starting phase-shifted to the right by $\pi/2$ .

Factoring out

b

from the argument reveals the phase shift

h=c/b

. The four parameters

a,b,h,k

completely determine the shape and position of the sinusoidal graph.

Example 2

hard

Write the equation of a cosine function with amplitude

4

, period

6

, phase shift right

1

, and vertical shift down

2

Example 3

medium

Identify amplitude, period, phase shift, and vertical shift of

y = -2\cos\!\left(\frac{x}{2} - \frac{\pi}{4}\right) + 3

Example 4

medium

A Ferris wheel of radius

10

m has its center

12

m above the ground and rotates once every

40

s. Write the height

h(t)

if a rider starts at the lowest point at

t=0

Example 5

medium

Identify the period of

y = \tan\!\left(\dfrac{x}{3}\right)

and the location of its first positive asymptote.

Example 6

hard

The temperature in a city oscillates sinusoidally between

54\,^\circ\text{F}

and

86\,^\circ\text{F}

over a

24

-hour day, with the high at

3

p.m. Write

T(t)

where

t

is hours past midnight.

Example 7

hard

Sketch reasoning: how many times does

y = \sin x

intersect

y = x/4

[-2\pi, 2\pi]

Example 8

challenge

A spring is pulled

6

cm below equilibrium and released, oscillating with period

0.4

s. Write a position function

y(t)

(cm, above equilibrium positive) ignoring damping.

Practice Problems

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

Example 1

easy

State the amplitude and period of each: (a)

y=\sin(4x)

, (b)

y=5\cos(x)

, (c)

y=-2\sin\!\left(\frac{x}{3}\right)

Example 2

medium

A sound wave has the equation

P(t)=0.002\sin(440\pi t)

(pressure in Pa, time in seconds). Find the frequency (Hz) and explain the connection to the period.

Example 3

easy

What is the period of

y = \sin x

Example 4

easy

What is the amplitude of

y = 3\sin x

Example 5

easy

What is the maximum value of

y = \cos x

Example 6

easy

What is the period of

y = \tan x

Example 7

easy

What is the amplitude of

y = -2\cos x

Example 8

easy

At what value of

x

[0, 2\pi)

does

y = \sin x

reach its maximum?

Example 9

easy

Where are the vertical asymptotes of

y = \tan x

closest to the origin?

Example 10

easy

What is the range of

y = \sin x

Example 11

medium

Find the period of

y = \sin(3x)

Example 12

medium

Find the amplitude and period of

y = 4\cos(2x)

Example 13

medium

Find the phase shift of

y = \sin(2x - \pi)

Example 14

medium

State the vertical shift and the resulting range of

y = 2\sin x + 5

Example 15

medium

How many complete cycles does

y = \cos(4x)

make on

[0, 2\pi]

Example 16

medium

Find the period of

y = \tan(2x)

Example 17

medium

Write a cosine function with amplitude 6, period

\pi

, and midline

y = -1

Example 18

medium

Find the period of

y = \sin\!\left(\frac{x}{2}\right)

Example 19

medium

Find the midline and range of

y = 3\cos x - 2

Example 20

challenge

The function

y = a\sin(bx) + d

oscillates between a high of 11 and a low of 3, completing one cycle every 8 units. Find

a

b

, and

d

Example 21

challenge

[0, 2\pi)

, how many solutions does

\sin(2x) = \frac{1}{2}

have?

Example 22

challenge

Explain why

y = \sin x

and

y = \cos x

are horizontal shifts of each other, and give the shift.

Example 23

easy

What is the minimum value of

y = \cos x

Example 24

easy

State the period of

y = \sin(\pi x)

Example 25

easy

What is the midline of

y = \sin x - 4

Example 26

easy

Does the graph of

y = -\sin x

open by reflection across the

x

-axis compared to

y = \sin x

Example 27

easy

State the range of

y = 5\cos x

Example 28

medium

Find the phase shift of

y = \cos(3x + \pi)

Example 29

medium

Find the amplitude and period of

y = -3\sin\!\left(\frac{\pi}{2}x\right)

Example 30

medium

How many complete cycles does

y = \sin(\pi x)

make on

[0, 6]

Example 31

medium

Write a sine equation with amplitude

2

, period

\pi

, and midline

y = 4

Example 32

medium

What are the maximum and minimum values of

y = 4\sin x - 2

Example 33

medium

Find the

x

-intercepts of

y = \sin(2x)

[0, 2\pi)

Example 34

medium

Compare the periods of

y = \sin(2x)

and

y = \sin(x/2)

: which is shorter?

Example 35

hard

Write a cosine equation oscillating between

-1

and

7

with period

\pi

and a minimum at

x = 0

Example 36

hard

[0, 2\pi)

, how many solutions does

\cos(3x) = 0

have?

Example 37

hard

Find all

x

[0, 2\pi)

where

2\sin x - 1 = 0

Example 38

challenge

Find all values of

b > 0

so that

y = \sin(bx)

has exactly

5

complete periods on

[0, 2\pi]

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Related Concepts

Trigonometric Functions Periodic Functions Function Transformation Inverse Trigonometric Functions

Background Knowledge

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

trigonometric functionsperiodic functionstransformation