slide 5.4- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley
TRANSCRIPT
Slide 5.4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
OBJECTIVES
Graphs of the Sine and Cosine Functions
Learn to define periodic functions.Learn to graph the sine and cosine functions.Learn to find the amplitude and period of sinusoidal functions.Learn to find the phase shift and graph sinusoidal functions of the forms y = a sin b(x – c) and y = a cos b(x – c).
SECTION 5.4
1
2
3
4
Slide 5.4- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION OF A PERIODIC FUNCTION
A function f is said to be periodic if there is a positive number p such that
f (x + p) = f (x) for every x in the domain of f.
The smallest value of p (if there is one) for which f (x + p) = f (x) is called the period of f. The graph of f over any interval of length p is called one cycle of the graph.
Slide 5.4- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
GRAPH OF THE SINE FUNCTION
If one were to make a table of values and plot
Slide 5.4- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
GRAPH OF THE COSINE FUNCTION
If one were to make a table of values and plot
Slide 5.4- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROPERTIES OF THE SINEAND COSINE FUNCTIONS
1. Period: 2π
2. Domain: (–∞, ∞)
3. Range: [–1, 1]
4. Odd: sin (–t) = –sin t
Sine Function Cosine Function
1. Period: 2π
2. Domain: (–∞, ∞)
3. Range: [–1, 1]
4. Even: cos (–t) = –cos t
Slide 5.4- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
CHANGING THE AMPLITUDE AND PERIOD OF THE SINE AND COSINE FUNCTIONS
The functions y = a sin bx and y = a cos bx (b > 0) have amplitude |a| and period
2b
.
If a > 0, the graphs of y = a sin bx and y = a cos bx are similar to the graphs of y = sin x and y = cos x , respectively, with two changes.
1. The range is [–a, a].
2. One cycle is completed over the
interval 0,2b
.
Slide 5.4- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
CHANGING THE AMPLITUDE AND PERIOD OF THE SINE AND COSINE FUNCTIONS
y asinbx
Slide 5.4- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
CHANGING THE AMPLITUDE AND PERIOD OF THE SINE AND COSINE FUNCTIONS
If a < 0, the graphs are the reflections of y = |a| sin bx and y = |a| cos bx, respectively, in the x-axis.
y a cosbx
Slide 5.4- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Graphing y = a cos bx
Graph over a one-period interval.
y 3cos1
2x
Solution
Amplitude is 3. 212
4 .Period is
Divide the period, 4π, into four quarters:0 to ππ to 2π2π to 3π3π to 4π
The five endpoints give the highest and lowest points and the x-intercepts of the graph.
Slide 5.4- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solution continued
Graphing y = a cos bx
Slide 5.4- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING y = a sin b(x – c) AND y = a cos b(x – c)
Step 1 Find the amplitude, period, and phase shift. amplitude = |a|
phase shift = c
period =2b
Step 2 The starting point for the cycle is x = c. The interval over which one complete
c,c 2b
.cycle occurs is
Slide 5.4- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING y = a sin b(x – c) AND y = a cos b(x – c)
Step 3 Divide the interval c,c 2b
into four
equal parts, each of length
This requires 51
4period 1
4
2b
.
points: a starting point c, c 1
4
2b
,
c 1
2
2b
, c 3
4
2b
, and c 2b
.
Slide 5.4- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING y = a sin b(x – c) AND y = a cos b(x – c)
Step 4 If a > 0, for y = a sin b(x – c) , sketch one cycle of the sine curve, starting at (c, 0),
c 2b
,a
,through the points
and c 2b
,0
.
c b
,0
,
c 32b
, a
,
Slide 5.4- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING y = a sin b(x – c) AND y = a cos b(x – c)
Step 4 continuedFor y = a cos b(x – c), sketch one cycle of the cosine curve, starting at (c, a),
c 2b
,0
,through the points
and c 2b
,a
.c b
, a
, c 32b
,0
,
Slide 5.4- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR GRAPHING y = a sin b(x – c) AND y = a cos b(x – c)
Step 4 continuedFor a < 0, reflect the graph of y = |a| sin b(x – c) or y = |a| cos b(x – c), in the x-axis.
Slide 5.4- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9 Graphing y = a sin b(x – c)
Graph over a one-period interval.
y 3sin 2x 2
Solution
Rewrite is as: y 3sin 2 x 4
Amplitude = 3 22
Period Phase shift 4
Starting point 4
One cycle4
,4
Slide 5.4- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9 Graphing y = a sin b(x – c)
Solution continued
1
4period
1
4
4starting point
4
2nd pt 4
1
4
23rd pt
4
1
2 3
4
4th pt 4
3
4 end pt
4
54
Slide 5.4- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9 Graphing y = a sin b(x – c)
Solution continuedy 3sin 2x
2
Slide 5.4- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 10 Graphing y = a cos b(x – c) + d
Graph over a one-period interval.
y cos2 x 2
3
Solution
Amplitude = 1 22
Period Phase shift 2
Starting point 2
One cycle 2
, 2
2
,2
Slide 5.4- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 10 Graphing y = a cos b(x – c) + d
Solution continued
1
4period
1
4
4starting point
2
2nd pt 2
1
4
4
3rd pt 2
1
2 0
4th pt 2
3
4
4end pt
2
2
Slide 5.4- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 10 Graphing y = a cos b(x – c) + d
Solution continued
Slide 5.4- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SIMPLE HARMONIC MOTION
An object whose position relative to an equilibrium position at time t can be described by either
y asint or y a cost 0
is said to be in simple harmonic motion.
Slide 5.4- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SIMPLE HARMONIC MOTION
The amplitude, |a| is the maximum distance the object attains from its equilibrium position.
2
,The period of the motion, is the time it
takes for the object to complete one full cycle.
The frequency of the motion is and gives the
number of cycles completed per unit time.
2
Slide 5.4- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 12Simple Harmonic Motion of a Ball Attached to a Spring
Suppose that a ball attached to a spring is pulled down 6 inches and released and the resulting simple harmonic motion has a period of 8 seconds. Write an equation for the ball ﾕ s simple harmonic motion.
SolutionChoose between y = a sin t or y = a cos t. For t = 0, y = a sin •0 = 0 and y = a cos •0 = a. Because we pulled the ball down in order to start, a is negative.
Slide 5.4- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 12Simple Harmonic Motion of a Ball Attached to a Spring
Solution continuedIf we start tracking the ball’s motion when we release it after pulling it down 6 inches, we should choose a = –6 and y = –6cos t. We have the form of the equation of motion:
y 6 cost
period 2
8, so 28
4
y 6 cos4
t.So the equation of the ball’s simple harmonic motion is