4.6 — Graphs of Sine and Cosine

Section 4.6

Graphs of Sine and Cosine

Objective: Understand periodic functions, amplitude, period, and transformations of sine and cosine graphs.

\[y = a\sin(b(x - c)) + d\]

Amplitude Period Phase Shift Vertical Translation Transformations

Setup

Label Your Graph Paper

X-axis: multiples of \(\frac{\pi}{2}\) from \(-2\pi\) to \(2\pi\) | Y-axis: integers from \(-2\) to \(2\)

Graph

Graph \(y = \sin x\)

\(x\)	\(\sin x\)
\(0\)	\(0\)
\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)
\(\frac{\pi}{2}\)	\(1\)
\(\frac{3\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)
\(\pi\)	\(0\)
\(\frac{5\pi}{4}\)	\(-\frac{\sqrt{2}}{2}\)
\(\frac{3\pi}{2}\)	\(-1\)
\(\frac{7\pi}{4}\)	\(-\frac{\sqrt{2}}{2}\)
\(2\pi\)	\(0\)

Click each cell to reveal its value and plot the point

Key Features

\(y = \sin x\)

Definition

Periodic Function

A function that returns to the same value at regular intervals.

Definition

Period

The least amount of space (degrees or radians) the function takes to complete one cycle.

Definition

Amplitude

Half the distance between the maximum and minimum values.

Graph

Graph \(y = \cos x\)

\(x\)	\(\cos x\)
\(0\)	\(1\)
\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)
\(\frac{\pi}{2}\)	\(0\)
\(\frac{3\pi}{4}\)	\(-\frac{\sqrt{2}}{2}\)
\(\pi\)	\(-1\)
\(\frac{5\pi}{4}\)	\(-\frac{\sqrt{2}}{2}\)
\(\frac{3\pi}{2}\)	\(0\)
\(\frac{7\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)
\(2\pi\)	\(1\)

Click each cell to reveal its value and plot the point

Key Features

\(y = \cos x\)

Properties

Amplitude & Period of \(y = \cos x\)

Comparison

Sine vs Cosine

\(\sin x\)

\(\cos x\)

Reference

Basic Characteristics of Sine and Cosine

	\(f(x) = \sin(x)\)	\(f(x) = \cos(x)\)
Domain	\((-\infty, \infty)\)
Range	\([-1, 1]\)
Period	\(2\pi\)
x-intercepts	\((n\pi, 0)\)	\((\frac{\pi}{2} + n\pi, 0)\)
y-intercept	\((0, 0)\)	\((0, 1)\)
Even/Odd	odd	even
Symmetry	about the origin	about the y-axis

Explore

Technology Exploration

Go to:

bit.ly/SineOnDesmos

Explore how changing parameters affects the graph.

Reference

General Sinusoidal Form

\[y = a\sin(b(x - c)) + d\]

a = Amplitude

Vertical stretch/shrink

b = Affects Period

Horizontal stretch/shrink. Period \(= \frac{2\pi}{|b|}\)

c = Phase Shift

Horizontal translation

d = Vertical Translation

Shifts the midline up or down

Example 1

Analyze \(y = 3\sin\!\left(\frac{x}{2}\right)\)

Identify the amplitude, period, phase shift, and vertical translation. Describe the transformations from \(y = \sin x\).

Identify Parameters

\[y = 3\sin\!\left(\tfrac{1}{2}\,x\right) \quad\Longrightarrow\quad a = 3,\; b = \tfrac{1}{2},\; c = 0,\; d = 0\] Vertical stretch by 3, horizontal stretch by 2

Amplitude

\[|a| = 3\]

Period

\[\text{Period} = \dfrac{2\pi}{|b|} = \dfrac{2\pi}{\frac{1}{2}} = 4\pi\]

Phase Shift & Vertical Translation

\[\text{Phase shift} = 0 \qquad \text{Vertical translation} = 0\]

\(y = \sin x\)

\(y = 3\sin(\frac{x}{2})\)

Example 2

Analyze \(y = \frac{1}{2}\sin\!\left(x - \frac{\pi}{3}\right)\)

Identify the amplitude, period, phase shift, and vertical translation. Describe the transformations from \(y = \sin x\).

Identify Parameters

\[a = \tfrac{1}{2},\; b = 1,\; c = \tfrac{\pi}{3},\; d = 0\] Vertical shrink by \(\frac{1}{2}\), shift right by \(\frac{\pi}{3}\)

Amplitude

\[|a| = \dfrac{1}{2}\]

Period

\[\text{Period} = \dfrac{2\pi}{|b|} = \dfrac{2\pi}{1} = 2\pi\]

Phase Shift & Vertical Translation

\[\text{Phase shift} = \dfrac{\pi}{3} \text{ to the right} \qquad \text{Vertical translation} = 0\]

\(y = \sin x\)

\(y = \frac{1}{2}\sin(x - \frac{\pi}{3})\)

Note

Phase Shift Convention

Our Convention

\[y = a\sin(b(x - c)) + d\]

Phase shift \(= c\)

Some Resources Use

\[y = a\sin(bx - c) + d\]

Phase shift \(= \dfrac{c}{b}\)

Note

These Are NOT the Same

Form 1

\[y = \sin(2x - \pi)\]

Rewrite as: \(y = \sin(2(x - \frac{\pi}{2}))\)

Phase shift \(= \dfrac{\pi}{2}\)

Form 2

\[y = \sin(2(x - \pi))\]

Already in correct form

Phase shift \(= \pi\)

Example 3

Analyze \(y = -3\cos(2\pi x + 4\pi)\)

Rewrite as \(y = -3\cos(2\pi(x + 2))\). Identify the amplitude, period, phase shift, and vertical translation.

Identify Parameters

\[a = -3,\; b = 2\pi,\; c = -2,\; d = 0\] Vertical stretch by 3, reflection over x-axis, horizontal shrink, shift left 2

Amplitude

\[|a| = 3 \quad\text{(reflected in the x-axis)}\]

Period

\[\text{Period} = \dfrac{2\pi}{|b|} = \dfrac{2\pi}{2\pi} = 1\]

Phase Shift

\[c = -2 \quad\Longrightarrow\quad \text{2 units to the left}\]

Vertical Translation

\[d = 0 \quad\Longrightarrow\quad \text{none}\]

\(y = \cos x\)

\(y = -3\cos(2\pi(x+2))\)

Example 4

Write an Equation from a Graph

The graph of a trig function is shown. (a) Find the amplitude, period, and phase shift. (b) The equation is \(y = a\sin(b(x - c))\). Find \(a\), \(b\), and \(c\).

Amplitude

\[|a| = 2 \quad\text{(reflected: } a = -2\text{)}\] Vertical stretch by 2, reflected over x-axis

Period

\[\text{Period} = 2\pi \quad\Longrightarrow\quad b = 1\]

Phase Shift

\[\text{Shift} = \dfrac{\pi}{2} \text{ to the left} \quad\Longrightarrow\quad c = -\dfrac{\pi}{2}\]

Equation

\[y = -2\sin\!\left(x + \dfrac{\pi}{2}\right)\]

What are your questions?

Assignment

Homework

Worksheet 1

Graphs of Sine and Cosine Amplitude and Period worksheet

Worksheet 2

Graphs of Sine and Cosine CIRCUIT worksheet