Trigonometric Function Graph: Functions and Their Modeling - Mathematics (Grade 11)

Understanding Trigonometric Function Graphs

Have you ever seen ocean waves? Their up-and-down movement forms patterns that repeat regularly. It turns out that these patterns can be modeled with trigonometric functions.

Before studying trigonometric function graphs, we need to understand angle measurement in radians. In daily life, we are accustomed to using degrees. However, in advanced mathematics, radians are more frequently used.

Converting Degrees and Radians

One complete rotation of a circle is $360^\circ$ or $2\pi$ radians. This relationship gives us conversion formulas:

180^\circ = \pi \text{ radians}

Conversion Examples

Converting degrees to radians:

90^\circ = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians}

Converting radians to degrees:

\frac{\pi}{6} \text{ radians} = \frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ

What are Amplitude and Period?

Before studying trigonometric function graphs, it's important to understand two key concepts: amplitude and period.

Amplitude

Amplitude is the maximum distance from the center line ( $x$ -axis) to the peak or trough of the graph. For functions $y = A \sin x$ or $y = A \cos x$ , the amplitude is $|A|$ .

Amplitude Concept

Amplitude determines the 'height' of the wave. Notice the distance from the

x

-axis to the peak.

Period

Period is the length of interval needed for one complete cycle. For functions $y = \sin(Bx)$ or $y = \cos(Bx)$ , the period is $\frac{2\pi}{|B|}$ .

Period Concept

Period is the horizontal distance for one complete wave.

General Formulas

For trigonometric functions in the form:

$y = A \sin(Bx)$ and $y = A \cos(Bx)$ :
$y = A \sin(Bx)$
$y = A \tan(Bx)$ :
$y = A \tan(Bx)$

Sine Function Graph

The function $y = \sin x$ is a periodic function with period $2\pi$ . This means its graph pattern repeats every $2\pi \text{ interval}$ .

Graph of

y = \sin x

Notice how the graph forms waves that repeat regularly.

Characteristics of $y = \sin x$ graph:

Period: $2\pi$ (graph repeats every $2\pi \text{ units}$ )
Amplitude: $1$ (maximum value minus minimum value, then divided by $2$ )
Domain: All real numbers
Range: $[-1, 1]$
x-intercepts: $x = n\pi$ where $n$ is an integer
Maximum value: $1$ at $x = \frac{\pi}{2} + 2n\pi$
Minimum value: $-1$ at $x = \frac{3\pi}{2} + 2n\pi$

Cosine Function Graph

The function $y = \cos x$ has a shape similar to sine, but shifted $\frac{\pi}{2}$ to the left.

Graph of

y = \cos x

Compare with the

\sin

graph. Notice the shift.

Characteristics of $y = \cos x$ graph:

Period: $2\pi$
Amplitude: $1$
Domain: All real numbers
Range: $[-1, 1]$
x-intercepts: $x = \frac{\pi}{2} + n\pi$
Maximum value: $1$ at $x = 2n\pi$
Minimum value: $-1$ at $x = \pi + 2n\pi$

Comparison of Sin and Cos

Comparison of

\sin

and

\cos

Graphs

Notice that

\cos x = \sin(x + \frac{\pi}{2})

Tangent Function Graph

The function $y = \tan x$ differs from $\sin$ and $\cos$ because it has vertical asymptotes.

Graph of

y = \tan x

Notice the dashed lines showing vertical asymptotes.

Characteristics of $y = \tan x$ graph:

Period: $\pi$ (shorter than $\sin$ and $\cos$ )
Amplitude: Undefined
Domain: $x \neq \frac{\pi}{2} + n\pi$
Range: All real numbers
Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$
x-intercepts: $x = n\pi$

Transformations of Trigonometric Functions

Amplitude Changes

The function $y = A \sin x$ changes the amplitude to $|A|$ .

Effect of Amplitude

Notice how the value of

A

affects the wave height.

Period Changes

The function $y = \sin(Bx)$ changes the period to $\frac{2\pi}{|B|}$ .

Effect of Period

The value of

B

affects how fast the function repeats.

Vertical and Horizontal Shifts

General form:

y = A \sin(B(x - C)) + D

$A$ : Amplitude
$B$ : Affects period ( $\text{period} = \frac{2\pi}{|B|}$ )
$C$ : Horizontal shift (phase)
$D$ : Vertical shift

Notice the horizontal and vertical shifts of the graph:

Complete Transformation

Graph of

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

shows all transformations.

Exercises

Convert the following angles:
- $120^\circ$ to radians
- $\frac{5\pi}{6}$ radians to degrees
Determine the period and amplitude of:
- $y = 3 \sin(2x)$
- $y = -2 \cos(\frac{x}{3})$
Sketch the graph of $y = 2 \sin(x + \frac{\pi}{3}) - 1$ . Determine:
- Amplitude
- Period
- Phase shift
- Vertical shift
If tidal height is modeled by $h(t) = 2 \sin(\frac{\pi t}{6}) + 5 \text{ meters}$ , where $t$ is measured in hours:
- What are the maximum and minimum water heights?
- What is the tidal period?
Determine the equation of a trigonometric function that has:
- Amplitude $3$
- Period $\pi$
- Shifted $\frac{\pi}{4}$ to the right
- Shifted $2 \text{ units}$ up

Answer Key

Angle conversion:
- $120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}$ radians
- $\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ$
Period and amplitude:
- $y = 3 \sin(2x)$ : amplitude $3$ , period $\frac{2\pi}{2} = \pi$
- $y = -2 \cos(\frac{x}{3})$ : amplitude $2$ , period $\frac{2\pi}{1/3} = 6\pi$
For $y = 2 \sin(x + \frac{\pi}{3}) - 1$ :
- Amplitude: $2$
- Period: $2\pi$
- Phase shift: $\frac{\pi}{3}$ to the left
- Vertical shift: $1 \text{ unit}$ down
For $h(t) = 2 \sin(\frac{\pi t}{6}) + 5$ :
- Maximum height: $5 + 2 = 7 \text{ meters}$
- Minimum height: $5 - 2 = 3 \text{ meters}$
- Period: $\frac{2\pi}{\pi/6} = 12 \text{ hours}$
Equation that satisfies the requirements:

$y = 3 \sin(2(x - \frac{\pi}{4})) + 2$

or

$y = 3 \sin(2x - \frac{\pi}{2}) + 2$

Understanding Trigonometric Function Graphs

Have you ever seen ocean waves? Their up-and-down movement forms patterns that repeat regularly. It turns out that these patterns can be modeled with trigonometric functions.

Converting Degrees and Radians

One complete rotation of a circle is $360^\circ$ or $2\pi$ radians. This relationship gives us conversion formulas:

180^\circ = \pi \text{ radians}

Conversion Examples

Converting degrees to radians:

90^\circ = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians}

Converting radians to degrees:

\frac{\pi}{6} \text{ radians} = \frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ

What are Amplitude and Period?

Before studying trigonometric function graphs, it's important to understand two key concepts: amplitude and period.

Amplitude

Amplitude is the maximum distance from the center line ( $x$ -axis) to the peak or trough of the graph. For functions $y = A \sin x$ or $y = A \cos x$ , the amplitude is $|A|$ .

Amplitude Concept

Amplitude determines the 'height' of the wave. Notice the distance from the

x

-axis to the peak.

Period

Period is the length of interval needed for one complete cycle. For functions $y = \sin(Bx)$ or $y = \cos(Bx)$ , the period is $\frac{2\pi}{|B|}$ .

Period Concept

Period is the horizontal distance for one complete wave.

General Formulas

For trigonometric functions in the form:

$y = A \sin(Bx)$ and $y = A \cos(Bx)$ :
$y = A \sin(Bx)$
$y = A \tan(Bx)$ :
$y = A \tan(Bx)$

Sine Function Graph

The function $y = \sin x$ is a periodic function with period $2\pi$ . This means its graph pattern repeats every $2\pi \text{ interval}$ .

Graph of

y = \sin x

Notice how the graph forms waves that repeat regularly.

Characteristics of $y = \sin x$ graph:

Period: $2\pi$ (graph repeats every $2\pi \text{ units}$ )
Amplitude: $1$ (maximum value minus minimum value, then divided by $2$ )
Domain: All real numbers
Range: $[-1, 1]$
x-intercepts: $x = n\pi$ where $n$ is an integer
Maximum value: $1$ at $x = \frac{\pi}{2} + 2n\pi$
Minimum value: $-1$ at $x = \frac{3\pi}{2} + 2n\pi$

Cosine Function Graph

The function $y = \cos x$ has a shape similar to sine, but shifted $\frac{\pi}{2}$ to the left.

Graph of

y = \cos x

Compare with the

\sin

graph. Notice the shift.

Characteristics of $y = \cos x$ graph:

Period: $2\pi$
Amplitude: $1$
Domain: All real numbers
Range: $[-1, 1]$
x-intercepts: $x = \frac{\pi}{2} + n\pi$
Maximum value: $1$ at $x = 2n\pi$
Minimum value: $-1$ at $x = \pi + 2n\pi$

Comparison of Sin and Cos

Comparison of

\sin

and

\cos

Graphs

Notice that

\cos x = \sin(x + \frac{\pi}{2})

Tangent Function Graph

The function $y = \tan x$ differs from $\sin$ and $\cos$ because it has vertical asymptotes.

Graph of

y = \tan x

Notice the dashed lines showing vertical asymptotes.

Characteristics of $y = \tan x$ graph:

Period: $\pi$ (shorter than $\sin$ and $\cos$ )
Amplitude: Undefined
Domain: $x \neq \frac{\pi}{2} + n\pi$
Range: All real numbers
Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$
x-intercepts: $x = n\pi$

Transformations of Trigonometric Functions

Amplitude Changes

The function $y = A \sin x$ changes the amplitude to $|A|$ .

Effect of Amplitude

Notice how the value of

A

affects the wave height.

Period Changes

The function $y = \sin(Bx)$ changes the period to $\frac{2\pi}{|B|}$ .

Effect of Period

The value of

B

affects how fast the function repeats.

Vertical and Horizontal Shifts

General form:

y = A \sin(B(x - C)) + D

$A$ : Amplitude
$B$ : Affects period ( $\text{period} = \frac{2\pi}{|B|}$ )
$C$ : Horizontal shift (phase)
$D$ : Vertical shift

Notice the horizontal and vertical shifts of the graph:

Complete Transformation

Graph of

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

shows all transformations.

Exercises

Convert the following angles:
- $120^\circ$ to radians
- $\frac{5\pi}{6}$ radians to degrees
Determine the period and amplitude of:
- $y = 3 \sin(2x)$
- $y = -2 \cos(\frac{x}{3})$
Sketch the graph of $y = 2 \sin(x + \frac{\pi}{3}) - 1$ . Determine:
- Amplitude
- Period
- Phase shift
- Vertical shift
If tidal height is modeled by $h(t) = 2 \sin(\frac{\pi t}{6}) + 5 \text{ meters}$ , where $t$ is measured in hours:
- What are the maximum and minimum water heights?
- What is the tidal period?
Determine the equation of a trigonometric function that has:
- Amplitude $3$
- Period $\pi$
- Shifted $\frac{\pi}{4}$ to the right
- Shifted $2 \text{ units}$ up

Answer Key

Angle conversion:
- $120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}$ radians
- $\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ$
Period and amplitude:
- $y = 3 \sin(2x)$ : amplitude $3$ , period $\frac{2\pi}{2} = \pi$
- $y = -2 \cos(\frac{x}{3})$ : amplitude $2$ , period $\frac{2\pi}{1/3} = 6\pi$
For $y = 2 \sin(x + \frac{\pi}{3}) - 1$ :
- Amplitude: $2$
- Period: $2\pi$
- Phase shift: $\frac{\pi}{3}$ to the left
- Vertical shift: $1 \text{ unit}$ down
For $h(t) = 2 \sin(\frac{\pi t}{6}) + 5$ :
- Maximum height: $5 + 2 = 7 \text{ meters}$
- Minimum height: $5 - 2 = 3 \text{ meters}$
- Period: $\frac{2\pi}{\pi/6} = 12 \text{ hours}$
Equation that satisfies the requirements:

$y = 3 \sin(2(x - \frac{\pi}{4})) + 2$

or

$y = 3 \sin(2x - \frac{\pi}{2}) + 2$