Graphing and Analyzing Trigonometric Functions: Amplitude, Period, and Phase Shift

Trigonometric Functions: Amplitude, Period, and Phase Shift

Introduction

Trigonometric functions such as sine and cosine are fundamental in Precalculus, especially for modeling periodic phenomena. Understanding how to determine and interpret the amplitude, period, and phase shift of these functions is essential for graphing and analyzing their behavior.

Basic Properties of Sine and Cosine Functions

General Form

The general form of a sine or cosine function is:

• Sine: $y = A \sin(Bx + C) + D$

• Cosine: $y = A \cos(Bx + C) + D$

Where:

• Amplitude ($|A|$): The maximum distance from the midline to the peak or trough.

• Period ($\frac{2\pi}{|B|}$): The length of one complete cycle.

• Phase Shift ($-\frac{C}{B}$): The horizontal shift of the graph.

• Vertical Shift ($D$): The upward or downward translation of the graph.

Topic 1: Determining Amplitude and Period

Amplitude

• The amplitude is the absolute value of the coefficient in front of the sine or cosine function.

• For $y = A \sin(x)$ or $y = A \cos(x)$, amplitude is $|A|$.

• Example: For $y = -6 \cos(x)$, amplitude is $|-6| = 6$.

Period

• The period is calculated as $\frac{2\pi}{|B|}$, where $B$ is the coefficient of $x$ inside the function.

• Example: For $y = 4 \sin(x)$, period is $\frac{2\pi}{1} = 2\pi$.

Topic 2: Graphing Sine and Cosine Functions

Steps for Graphing

1. Identify amplitude, period, phase shift, and vertical shift.

2. Mark the midline (if $D \neq 0$).

3. Divide the period into four equal intervals to plot key points (maximum, zero, minimum, zero, maximum).

4. Apply phase shift and vertical shift as needed.

Example Problems

• Given: $y = -8 \sin\left(\frac{1}{2}x\right)$

• Amplitude: $|-8| = 8$

• Period: $\frac{2\pi}{1/2} = 4\pi$

• Given: $y = 2 \cos(2x - \pi)$

• Amplitude: $|2| = 2$

• Period: $\frac{2\pi}{2} = \pi$

• Phase Shift: $\frac{\pi}{2}$ to the right

Topic 3: Phase Shift and Vertical Shift

Phase Shift

• Calculated as $-\frac{C}{B}$ from $y = A \sin(Bx + C)$ or $y = A \cos(Bx + C)$.

• Example: $y = -8 \sin\left(\frac{1}{3}x + \frac{\pi}{2}\right)$ Phase shift: $-\frac{\pi/2}{1/3} = -\frac{3\pi}{2}$ (to the left)

Vertical Shift

• Given by $D$ in $y = A \sin(Bx + C) + D$ or $y = A \cos(Bx + C) + D$.

• Example: $y = 4 \cos(x) + 2$ has a vertical shift of 2 units up.

Topic 4: Writing Equations from Graphs

To write an equation for a given sine or cosine graph:

1. Determine the amplitude (distance from midline to peak).

2. Find the period (distance between repeating points).

3. Identify any phase shift (horizontal translation).

4. Check for vertical shift (midline position).

5. Decide if the function is sine or cosine based on the starting point and shape.

Example Table: Properties of Trigonometric Functions

Function	Amplitude	Period	Phase Shift	Vertical Shift
$y = -6 \cos(x)$	6	$2\pi$	0	0
$y = 4 \sin(x)$	4	$2\pi$	0	0
$y = -8 \sin\left(\frac{1}{2}x\right)$	8	$4\pi$	0	0
$y = 2 \cos(2x - \pi)$	2	$\pi$	$\frac{\pi}{2}$ right	0
$y = 4 \cos(x) + 2$	4	$2\pi$	0	2

Topic 5: Applications and Practice

• Graphing trigonometric functions is essential for modeling waves, sound, and periodic motion.

• Practice involves identifying key properties and sketching accurate graphs over specified intervals.

• Writing equations from graphs requires careful analysis of amplitude, period, phase, and vertical shifts.

Additional info:

• Some problems in the file ask for both sine and cosine equations for the same graph, reinforcing the concept that many periodic graphs can be represented by either function with appropriate phase shifts.