CHAPTER 44-5 GRAPHS OF SINE AND COSINE FUNCTIONS

OBJECTIVES

Sketch the graphs of basic sine and cosine functions.

• Use amplitude and period to help sketch the graphs of sine and cosine functions.

• Sketch translations of the graphs of sine and cosine functions.

• Use sine and cosine functions to model real-life data.

BASIC SINE AND COSINE FUNCTIONS

The Sine Curve y = a sin t

The sine curve occurs naturally when we are examining waves. When waves have more energy, they go up and down more vigorously. We say they have greater amplitude.

Let's investigate the shape of the curve y = a sin t and see what the concept of "amplitude" means.

BASIC SINE AND COSINE FUNCTIONS

The black portion of the graph represents one period of the function and is called one cycle of the sine curve. The domain of the sine and cosine functions is the set of all real numbers. The range of each function is the interval [–1, 1]. Each function has a period of 2 .

AMPLITUDE

The "a" in the expression y = a sin x represents the amplitude of the graph. It is an indication of how much energy the wave contains. The amplitude is the distance from the "resting" position (otherwise known as the mean value or average value) of the curve. In the interactive above, the amplitude can be varied from 10  to 100  units.

Amplitude is always a positive quantity. We could write this using absolute value signs. For the curve y = a sin x, amplitude =|a| 

PERIOD

The variable b in both of the graph types

y = a sin bx

y = a cos bx

affects the period (or wavelength) of the graph. The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again.

The relationship between b  and the period is given by:Period=2π/ b   

Note: As b gets larger, the period decreases.

PHASE SHIFTS

y= a sin(bx + c)andy = a cos(bx + c)Both b and c in these graphs affect the phase shift (or displacement), given by:Phase shift=−c/ b    The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. There is nothing magic about this formula. We are just solving the expression in brackets for zero; bx+c=0  .

KEYS TO GRAPH SINE AND COSINE FUNCTIONS

Five key points in one period of each graph: the intercepts, maximum points, and minimum points

EXAMPLE#1

Find the period and amplitude of the function y  = 10 sin 5x  .

EXAMPLE#2

What is the period and amplitude of the function y  = cos 12x ?

EXAMPLE#3

What is the period and amplitude of the function y  = (- 5/ 6) sin 3x?

STUDENT GUIDED PRACTICE

Do problems 11-14 in your book page 299

EXAMPLE

Sketch the graph of y = 2 sin x on the interval [–

, 4

].

Solution: Note that y = 2 sin x indicates that the y-values for the key points will have twice the magnitude of those on the graph of y = sin x.

Divide the period 2

into four equal parts to get the key points for y = 2 sin x.

Intercept Maximum Intercept Minimum Intercept

and

= 2(sin x)

7

Example 1 – Solution

By connecting these key points with a smooth curve and extending the curve in both directions over the interval [– , 4 ], you obtain the graph shown in Figure 4.50.

c

SOLUTION

EXAMPLE

Sketch the graph of

Solution:

The amplitude is 1. Moreover, because b = 1/2 , the period is 4.

Now, divide the period-interval [0, 4

] into four equal parts

with the values

, 2

, and 3

to obtain the key points on the

graph.

Intercept Maximum Intercept Minimum Intercept (0, 0), ( , 1), (2 , 0), (3 , –1), and (4 , 0)

SOLUTION

Graphs

STUDENT GUIDED PRACTICE

Do problems 43-45 in your book page 300

TRANSLATIONS OF SINE AND COSINE GRAPHS

The constant c in the general equations

y = a sin(bx – c) and y = a cos(bx – c)

creates a horizontal translation (shift) of the basic sine and cosine curves.

EXAMPLE

Sketch the graph of y = –3 cos(2

x + 4

).

WORKSHEET

Do the worksheet

CLOSURE

Today we learned about the graphs of sine and cosine