4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the...
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Transcript of 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the...
![Page 1: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/1.jpg)
4.5 Graphs of Sine and Cosine
Functions
![Page 2: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/2.jpg)
In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are positive constants and x is in radian measure. The graphs of all sine and cosine functions are related to the graphs of y = sin x and y = cos xwhich are shown below.
y = sin x
y = cos x
![Page 3: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/3.jpg)
x
Sin x
Cos x
Fill in the chart.
2
2
3 20
These will be key points on the graphs of y = sin x and y = cos x.
![Page 4: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/4.jpg)
6. The cycle repeats itself indefinitely in both directions of the x-axis.
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
1. The domain is the set of real numbers.
5. Each function cycles through all the values of the range over an x-interval of .2
2. The range is the set of y values such that . 11 y
![Page 5: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/5.jpg)
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.
0-1010sin x
0x2
2
32
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.
y
2
3
2
22
32
2
5
1
1
x
y = sin x
![Page 6: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/6.jpg)
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.These are the maximum points, the minimum points, and the intercepts.
10-101cos x
0x2
2
32
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.
y
2
3
2
22
32
2
5
1
1
x
y = cos x
![Page 7: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/7.jpg)
Before sketching a graph, you need to know:
• Amplitude – Constant that gives vertical stretch or shrink.
• Period –
• Interval – Divide period by 4
• Critical points – You need 5.(max., min., intercepts.)
.2
b
![Page 8: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/8.jpg)
Amplitudes and PeriodsThe graph of y = A sin Bx has
amplitude = | A|period =
The graph of y = A sin Bx hasamplitude = | A|period =
To get your critical points (max, min, and intercepts) just
take your period and divide by 4.
Example: xy cos3
2
1
2Period
24
2
4
Period
2 ,
2
3 , ,
2 0,at come
willpoints critical So
2
2
2
2
B
2
Interval
![Page 9: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/9.jpg)
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically.If 0 < |a| < 1, the amplitude shrinks the graph vertically.If a < 0, the graph is reflected in the x-axis.
2
32
4
y
x
4
2
y = – 4 sin xreflection of y = 4 sin x y = 4 sin x
y = 2sin x
2
1y = sin x
y = sin x
Notice that since
all these graphs
have B=1, so the
period doesn’t
change.
![Page 10: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/10.jpg)
y
x
2
sin xy period: 2 2sin xy
period:
The period of a function is the x interval needed for the function to complete one cycle.
For b 0, the period of y = a sin bx is .b
2
For b 0, the period of y = a cos bx is also .b
2
If 0 < b < 1, the graph of the function is stretched horizontally.
If b > 1, the graph of the function is shrunk horizontally.
y
x 2 3 4
cos xy period: 2
2
1cos xy
period: 4
![Page 11: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/11.jpg)
y
1
123
2
x 32 4
Example 1: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
maxx-intminx-intmax
30-303y = 3 cos x
20x2
2
3
(0, 3)
2
3( , 0)( , 0)
2
2( , 3)
( , –3)
![Page 12: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/12.jpg)
Determine the amplitude of y = 1/2 sin x. Then graph y = sin x and y = 1/2 sin x for 0 < x < 2.
Example 2
![Page 13: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/13.jpg)
2
3x
y
2˝
˝
-1
1
y = sin x
y = 1/2sinx
2
2
2
1
2
1
![Page 14: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/14.jpg)
Example 3.
2sin
xyofgraphtheSketch
![Page 15: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/15.jpg)
Example 3.
2sin
xyofgraphtheSketch
![Page 16: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/16.jpg)
For the equations y = a sin(bx-c)+d and y = a cos(bx-c)+d
• a represents the amplitude. This constant acts as a scaling factor – a vertical stretch or shrink of the original function.
• Amplitude =• The period is the sin/cos curve making one complete
cycle. • Period =• c makes a horizontal shift.• d makes a vertical shift.• The left and right endpoints of a one-cycle interval can
be determined by solving the equations bx-c=0 and bx-c=
.a
.2
b
.2
![Page 17: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/17.jpg)
Example 4.
3sin
2
1
xyofgraphtheSketch
![Page 18: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/18.jpg)
Example 4
.3
sin2
1
xyofgraphtheSketch
![Page 19: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/19.jpg)
Example 6
.2cos32 xyofgraphtheSketch
![Page 20: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/20.jpg)
Example 6
.2cos32 xyofgraphtheSketch
![Page 21: 4.5 Graphs of Sine and Cosine Functions. In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cc15503460f94988439/html5/thumbnails/21.jpg)
Tides
• Throughout the day, the depth of the water at the end of a dock in Bangor, Washington varies with the tides. The tables shows the depths (in feet) at various times during the morning.
(a) Use a trig function to model the data.
(b) A boat needs at least 10 feet of water to moor at the dock. During what times in the evening can it safely dock?
Time Depth, y
Midnight 3.1
2 a.m. 7.8
4 a.m 11.3
6 a.m. 10.9
8 a.m. 6.6
10 a.m. 1.7
Noon 0.9cos( )y a bt c d
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Homework
• Page 304-305
• 3-13 odd, 15-21 odd, 31 – 34 all, 47-57 odd, 65, 67