Trigonometric Function Graph

Repeating Motion and Angle Measurement

Ocean waves rise and fall in a repeating pattern. Trigonometric functions can model this kind of periodic motion.

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:

Conversion Examples

Converting degrees to radians:

Converting radians to degrees:

Amplitude and Period of a Sine Graph

Two quantities describe the basic shape of a trigonometric graph: amplitude and period. The amplitude tells you how far the curve rises above and falls below its middle line, and the period tells you how much horizontal distance one full repetition needs.

Amplitude

Amplitude is the maximum vertical distance from a sinusoid's midline to a peak or trough. In the unshifted functions $y = A \sin x$ and $y = A \cos x$, that midline is the $x$-axis and the amplitude is $|A|$.

Period

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

Amplitude and Period from the Coefficients

In the form $y = A \sin(Bx)$, the factor $A$ sets the amplitude and the factor $B$ sets the period. The formulas below give both quantities from those two coefficients. Sine and cosine keep a defined amplitude, while tangent has none because its values are unbounded.

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

  $$y = A \sin(Bx)$$

  $$y = A \cos(Bx)$$

• $y = A \tan(Bx)$:

  $$y = A \tan(Bx)$$

  $$\text{Amplitude} = \text{undefined}$$

  $$\text{Period} = \frac{\pi}{|B|}$$

Sine Function Graph

The function $y = \sin x$ is periodic with period $2\pi$. Its graph pattern therefore repeats over every interval of length $2\pi$.

Graph characteristics: $y = \sin x$

• 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: $[-1, 1]$
• Intercepts on the $x$-axis: $x = n\pi$ 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 characteristics: $y = \cos x$

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

Comparison of Sin and Cos

Both curves repeat after a full turn and differ by a horizontal shift. Compare the peaks and the zero crossings of the two curves in the same view.

Tangent Function Graph

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

Graph characteristics: $y = \tan x$

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

Transformations of Trigonometric Functions

Each parameter acts at a different place in the equation, so one feature of the curve changes at a time. The examples show that effect on a graph.

Amplitude, period, and phase shift each change the curve in its own way.

Amplitude Changes

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

Period Changes

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

Vertical and Horizontal Shifts

General form:

• $A$: 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:

Exercises

The five problems below apply the period, amplitude, and shift rules to graphs you sketch yourself. Work each one before the answer key, then check how your sketches and equations compare with the recorded answers.

1. Convert the following angles:

   • $120^\circ$ to radians
   • $\frac{5\pi}{6}$ radians to degrees

2. Determine the period and amplitude of:

   • $y = 3 \sin(2x)$
   • $y = -2 \cos(\frac{x}{3})$

3. Sketch the graph of $y = 2 \sin(x + \frac{\pi}{3}) - 1$. Determine:

   • Amplitude
   • Period
   • Phase shift
   • Vertical shift

4. 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?

5. 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

1. Angle conversion:

   • $120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}$ radians
   • $\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ$

2. Period and amplitude:

   • $y = 3 \sin(2x)$: amplitude $3$, period $\frac{2\pi}{2} = \pi$
   • $y = -2 \cos(\frac{x}{3})$: amplitude $2$, period $\frac{2\pi}{1/3} = 6\pi$

3. 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

4. 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: $\frac{2\pi}{\pi/6} = 12 \text{ hours}$

5. Equation that satisfies the requirements:

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

   or

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