Graphs of Trigonometric Functions

Understanding and sketching sin, cos, and tan graphs with transformations

SPEC 5.3 • TRIGONOMETRY

Learning Objectives

By the end of this lesson

Sketch graphs of

y = sin x

,

y = cos x

, and

y = tan x

Understand periodicity, maximum/minimum values, and asymptotes

Apply transformations to trigonometric graphs

Sketch transformed trigonometric graphs

Determine equations from given graphs

Prior Knowledge

Basic trigonometric ratios, radian measure, and graph transformations from earlier topics.

The Graph of y = sin x

Key Features

Period:

2 π

(360°)

Maximum: 1

Minimum: −1

Range:

- 1 \leq y \leq 1

Key Points

sin 0 = sin 0° = 0

sin \frac{π}{2} = sin 90° = 1

sin π = sin 180° = 0

sin \frac{3 π}{2} = sin 270° = - 1

The Graph of y = cos x

Key Features

Period:

2 π

(360°)

Maximum: 1

Minimum: −1

Range:

- 1 \leq y \leq 1

Key Points

cos 0 = cos 0° = 1

cos \frac{π}{2} = cos 90° = 0

cos π = cos 180° = - 1

cos \frac{3 π}{2} = cos 270° = 0

Comparing sin x and cos x

The sine and cosine graphs have the same shape but are shifted horizontally:

Key Relationship

cos x = sin (x + \frac{π}{2})

The cosine graph is the sine graph shifted

\frac{π}{2}

to the left.

Symmetry

y = sin x

is an odd function:

sin (- x) = - sin x

(rotational symmetry about origin)

y = cos x

is an even function:

cos (- x) = cos x

(reflective symmetry in y-axis)

The Graph of y = tan x

Key Features

Period:

π

(180°)

No maximum or minimum

Range: All real

y

Asymptotes

Vertical asymptotes at:

x = \pm \frac{π}{2}, \pm \frac{3 π}{2}, ...

(i.e.,

\pm 90°, \pm 270°, ...

)

Key Properties Summary

Property

y = sin x

y = cos x

y = tan x

Period

2 π

(360°)

2 π

(360°)

π

(180°)

Maximum11None

Minimum−1−1None

Range

[- 1, 1]

[- 1, 1]

R

SymmetryOddEvenOdd

Passes through

(0, 0)

(0, 1)

(0, 0)

Transformations of Trig Graphs

The transformations you have learned can be applied to trigonometric graphs. For

y = sin x

:

TransformationEquationEffect on Graph

Vertical stretch (×a)

y = a sin x

Stretched by factor

a

in y-direction; max =

a

, min =

- a

Horizontal stretch (×

\frac{1}{a}

)

y = sin (a x)

Compressed by factor

\frac{1}{a}

; new period =

\frac{2 π}{a}

Translation left by

a

y = sin (x + a)

Shifted

a

units left

Translation up by

a

y = sin x + a

Shifted

a

units up; midline at

y = a

Reflection in x-axis

y = - sin x

Reflected in x-axis

Reflection in y-axis

y = sin (- x)

Reflected in y-axis

Remember

These same transformations apply to

y = cos x

and

y = tan x

. For tan, the new period is

\frac{π}{a}

.

01Worked Example

Sketch the graph of $y = 3 sin (2 x)$ for $0 \leq x \leq 2 π$ .

State the maximum value, minimum value, and period.

1

Identify transformations: vertical stretch ×3, horizontal compression ×½

2

Maximum value = 3, Minimum value = −3

3

Period =

\frac{2 π}{2} = π

(180°)

4

Key points: Starts at (0, 0), max at

(\frac{π}{4}, 3)

, zero at

(\frac{π}{2}, 0)

, min at

(\frac{3 π}{4}, - 3)

, completes cycle at

(π, 0)

5

Graph completes 2 full cycles between 0 and

2 π

AnswerMax = 3, Min = −3, Period =

π

(180°)

Worked Example02

Sketch the graph of $y = cos (x - \frac{π}{3}) + 1$ for $0 \leq x \leq 2 π$ .

1

Identify transformations: Phase shift

\frac{π}{3}

(60°) right, vertical shift 1 up

2

Max = 2, Min = 0, Period =

2 π

(360°)

3

Midline:

y = 1

, Range:

[0, 2]

4

Maximum at

x = \frac{π}{3}

(60°) where

cos = 1

, giving point

(\frac{π}{3}, 2)

5

Minimum at

x = \frac{π}{3} + π = \frac{4 π}{3}

(240°), giving point

(\frac{4 π}{3}, 0)

AnswerCosine curve shifted

\frac{π}{3}

right and 1 unit up, oscillating between 0 and 2

Worked Example03

A graph has maximum value 4, minimum value −4, period $π$ (180°), and passes through the origin with positive gradient. Find the equation in the form $y = a sin (b x)$ .

1

Max = 4, Min = −4, so

a = \frac{4 - ( - 4 )}{2} = 4

2

Period =

π

, so

\frac{2 π}{b} = π

, giving

b = 2

3

Check:

y = 4 sin (2 x)

passes through origin ✓

4

Check maximum:

sin (2 x) = 1

when

2 x = \frac{π}{2}

, so

x = \frac{π}{4}

(45°) ✓

Answer

y = 4 sin (2 x)

04Worked Example

Sketch the graph of $y = 2 tan (x - \frac{π}{4}) + 1$ for $0 \leq x \leq 2 π$ .

State the period and the equations of the asymptotes in this interval.

1

Identify transformations: vertical stretch ×2, translation

\frac{π}{4}

(45°) right, translation 1 up

2

Period =

π

(180°) — unchanged as no horizontal stretch

3

Original asymptotes at

x = \frac{π}{2}, \frac{3 π}{2}

shift right by

\frac{π}{4}

4

New asymptotes:

x = \frac{π}{2} + \frac{π}{4} = \frac{3 π}{4}

(135°) and

x = \frac{3 π}{2} + \frac{π}{4} = \frac{7 π}{4}

(315°)

5

Graph passes through

(\frac{π}{4}, 1)

since

tan 0 = 0

, then ×2 and +1 gives 1

AnswerPeriod =

π

(180°); Asymptotes:

x = \frac{3 π}{4}

(135°),

x = \frac{7 π}{4}

(315°)

P

Practice 1

Sketch the graph of $y = 2 sin (3 x) - 1$ for $0 \leq x \leq 2 π$ .

State: (a) the maximum value, (b) the minimum value, (c) the period.

a)

Maximum value = 2 − 1 = 1

b)

Minimum value = −2 − 1 = −3

c)

Period =

\frac{2 π}{3}

(120°)

•

Graph completes 3 full cycles between 0 and

2 π

P

Practice 2

A trigonometric graph has the equation $y = a cos (b x) + d$ .

The graph has maximum value 5, minimum value −1, and period

\frac{2 π}{3}

(120°).

Find the values of

a

,

b

, and

d

.

1

Vertical stretch:

a = \frac{max - min}{2} = \frac{5 - ( - 1 )}{2} = 3

2

Midline:

d = \frac{max + min}{2} = \frac{5 + ( - 1 )}{2} = 2

3

Period:

\frac{2 π}{b} = \frac{2 π}{3}

, so

b = 3

Answer

a = 3

,

b = 3

,

d = 2