5.3 Trigonometric Graphs

5.3.1 Learning Objectives

By the end of this section, you should be able to:

graph sine and cosine functions;
identify amplitude, period, phase shift, and midline;
write equations from graph features;
determine domains, ranges, and intercepts; and
use Desmos to verify sinusoidal graphs.

5.3.2 Basic Sine and Cosine Graphs

Both \(y=\sin x\) and \(y=\cos x\) have domain \(\mathbb R\), range \([-1,1]\), and period \(2\pi\). Sine begins at \(0\), while cosine begins at its maximum \(1\).

Example 5.9 Plotting One Sine Cycle

Give the five key points for \(y=\sin x\) on \([0,2\pi]\).

Solution

\[ \boxed{(0,0),\ (\pi/2,1),\ (\pi,0),\ (3\pi/2,-1),\ (2\pi,0)} \]

Exercise 5.9 Give the five key points for \(y=\cos x\) on \([0,2\pi]\).

5.3.3 Amplitude, Period, and Midline

For

\[ y=A\sin(Bx)+D \quad\text{or}\quad y=A\cos(Bx)+D, \]

the amplitude is \(|A|\), the period is \(2\pi/|B|\), and the midline is \(y=D\). The range is \([D-|A|,D+|A|]\).

Example 5.10 Reading Sinusoidal Features

Find the amplitude, period, midline, and range of \(y=-3\cos(2x)+1\).

Solution

\[ \boxed{\text{Amplitude }3,\quad \text{period }\pi,\quad \text{midline }y=1,\quad \text{range }[-2,4]} \]

Exercise 5.10 Find the amplitude and period of \(y=4\sin(x/3)\).

5.3.4 Phase Shifts

For

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

the graph shifts horizontally by \(C\). A convenient cycle runs from \(x=C\) to \(x=C+2\pi/|B|\), divided into four equal intervals.

Example 5.11 Graphing a Shifted Cosine

Describe \(y=2\cos(x-\pi/3)-1\).

Solution

The amplitude is \(2\), the period is \(2\pi\), the phase shift is \(\pi/3\) right, and the midline is \(y=-1\).

\[ \boxed{\text{Amplitude }2,\quad \text{period }2\pi,\quad \text{right }\pi/3,\quad y=-1} \]

Exercise 5.11 State the phase shift of \(y=\sin(3(x+\pi/4))\).

5.3.5 Writing a Sinusoidal Equation

From a graph, calculate the midline as the average of the maximum and minimum, and the amplitude as half their difference. Measure the horizontal length of one cycle to obtain the period.

Example 5.12 Building a Cosine Function

A sinusoid has maximum \(7\), minimum \(1\), period \(4\pi\), and a maximum at \(x=0\). Find an equation.

Solution

The midline is \(4\), the amplitude is \(3\), and \(B=2\pi/(4\pi)=1/2\). A cosine model is

\[ \boxed{y=3\cos(x/2)+4}. \]

Exercise 5.12 A sine graph has amplitude \(5\), period \(\pi\), and midline \(y=-2\), with no phase shift. Write an equation.

5.3.6 Conceptual Takeaways

Sine and cosine repeat every \(2\pi\).
Amplitude measures vertical distance from the midline.
The coefficient of \(x\) controls period.
Phase and vertical shifts reposition a cycle.
Five equally spaced key points describe one sinusoidal cycle.

5.3.7 Skills You Should Be Able to Do

Graph basic sine and cosine.
Calculate amplitude, period, midline, phase shift, and range.
Plot one transformed cycle.
Write a sinusoidal equation from features.
Verify a graph using Desmos.

5.3.8 Practice Problems with Solutions

Find the amplitude and period of \(y=6\sin(4x)\).

Show Solution

\[ \boxed{\text{Amplitude }6,\quad \text{period }\pi/2}. \]
Find the midline and range of \(y=2\cos x-5\).

Show Solution

The midline is \(y=-5\), and the graph extends \(2\) units above and below:

\[ \boxed{\text{Range }[-7,-3]}. \]
Find the period of \(y=\sin(3x/2)\).

Show Solution

\[ \boxed{\frac{2\pi}{3/2}=\frac{4\pi}{3}}. \]
Describe \(y=-4\sin(x+\pi)+2\).

Show Solution

It has amplitude \(4\), period \(2\pi\), reflection across the \(x\)-axis, shift left \(\pi\), and midline \(y=2\).

\[ \boxed{|A|=4,\quad T=2\pi,\quad C=-\pi,\quad D=2} \]
Write a cosine equation with amplitude \(2\), period \(3\pi\), midline \(y=1\), and a maximum at \(x=0\).

Show Solution

Since \(B=2\pi/(3\pi)=2/3\),

\[ \boxed{y=2\cos(2x/3)+1}. \]
Use Desmos to graph \(y=3\sin(2(x-\pi/4))+1\). State its maximum and minimum values.

Show Solution

The midline is \(1\) and amplitude is \(3\):

\[ \boxed{\text{maximum }4,\quad \text{minimum }-2}. \]