Transcript 4 5 Sin Cos Graphs

Objectives:
1. To graph sine and
cosine functions as
transformations on a
parent function
2. To find the period,
amplitude, and
phase shift of sine or
cosine curves
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Assignment:
P. 328: 1-14 S
P. 328: 15-26 S
P. 329: 27-34 S
P. 329: 35-56 S
P. 329: 63-70 S
P. 330: 73, 76, 77
P. 331: 81-83, 86
You will be able to graph sine and
cosine functions as transformations
on a parent function
Use SRT transformations to graph the function.
1
y2
x 1  3
2
1
y  2  x  2  3
2
S: 2x, 2y
R: None
T: R2, D3
The sine and cosine ratios can be defined as a
function of x, where x is a real number.
f ( x)  d  a sin  bx  c 
g ( x)  d  a cos  bx  c 
x is measured in radians if it refers to an
angle, but it doesn’t have to represent an
angle
The sine and cosine ratios can be defined as a
function of x, where x is a real number.
y  d  a sin  bx  c 
y  d  a cos  bx  c 
These functions can be used to
mathematically describe repetitive behavior:
blood pressure, sound, tide heights, vibrating
strings, OCD (just kidding about that one)
The sine and cosine ratios can be defined as a
function of x, where x is a real number.
y  d  a sin  bx  c 
y  d  a cos  bx  c 
Just like other functions, we frequently graph
each of these as a sine or cosine curve where
the parameters a, b, c, and d all do
something extraordinary!
First, though, we need to learn how to graph the
sine and cosine parent functions.
f ( x)  sin x
g ( x)  cos x
Guess what? Graphing these uses the unit
circle.
• For both functions the x will be the angle
measure in radians.
First, though, we need to learn how to graph the
sine and cosine parent functions.
f ( x)  sin x
g ( x)  cos x
For sine, f (x) is the y-coordinate of the point
on the unit circle corresponding to the angle
measure.
First, though, we need to learn how to graph the
sine and cosine parent functions.
f ( x)  sin x
g ( x)  cos x
For cosine, g(x) is the x-coordinate of the
point on the unit circle corresponding to
angle measure.
1. Start by filling in the needed coordinates on the unit
circle: radian measures, y-coordinates.
2. On the coordinate plane, the y-axis increases by
tenths. The x-axis increases by the angle measures
on your unit circle: 0, π/6, π/4, π/3, π/2, etc.
3. What’s the y-coordinate of x = 0 radians? Plot this
as a point on the coordinate plane:
(angle measure, y-coordinate)
4. What’s the y-coordinate of x = π/6 radians? Plot
this as a point on the coordinate plane:
(angle measure, y-coordinate)
Notice that the height of the triangle at π/6 (y-value) is
the same height of this point above the x-axis.
5. Continue this process all the way around the unit
circle. This will complete one cycle or period of the
sine curve.
f ( x)  sin x
Sine = Snake
For this cycle, notice
these 5 key
points: the 3
intercepts and the
min/max points.
These are all you
really need to plot
your sine curve.
6. Can’t we measure our angles over 2π? Well, of
course. So our sine curve should continue past 2π.
To continue the curve, either use more points from
the unit circle or use the 5 key points.
7. Can’t we measure our angles under 0? Well, of
course, these are just negative angle. To continue
the curve in the negative x direction, either use
more points from the unit circle or use the 5 key
points.
f ( x)  sin x
Sine = Snake
Domain: All Real #s
Range: [−1, 1]
Period: 2π
Looks Like: A Snake
Zeros: {…, −2π, −π, 0,
π, 2π, …}
Symmetry: Origin
1. Start by filling in the needed coordinates on the unit
circle: radian measures, x-coordinates.
2. On the coordinate plane, the y-axis increases by
tenths. The x-axis increases by the angle measures
on your unit circle: 0, π/6, π/4, π/3, π/2, etc.
3. What’s the x-coordinate of 0 radians? Plot this as a
point on the coordinate plane:
(angle measure, x-coordinate)
Don’t let the fact that we’re using two x-coordinates
here, one of which is a y-coordinate, confuse you!
4. What’s the x-coordinate of π/6 radians? Plot this as
a point on the coordinate plane:
(angle measure, x-coordinate)
Notice that the base of the triangle at π/6 (x-value) is
the same height of this point above the x-axis.
5. Continue this process all the way around the unit
circle. This will complete one cycle or period of the
cosine curve.
g ( x)  cos x
Cosine = Cup
For this cycle, notice
these 5 key
points: the 3
intercepts and the
min/max points.
These are all you
really need to plot
your cosine curve.
6. Can’t we measure our angles over 2π? Well, of
course. So our cosine curve should continue past
2π. To continue the curve, either use more points
from the unit circle or use the 5 key points.
7. Can’t we measure our angles under 0? Well, of
course, these are just negative angle. To continue
the curve in the negative x direction, either use
more points from the unit circle or use the 5 key
points.
g ( x)  cos x
Cosine = Cup
Domain: All Real #s
Range: [−1, 1]
Period: 2π
Looks Like: A Cup
Zeros: {…, −3π/2,
−π/2, π/2, 3π/2,
…}
Symmetry: y-axis
y  d  a sin  bx  c 
Since you are so clever, you
probably already know
what each of the
parameters do to a sine or
cosine curve. But since we
have a bit of time left in
the period, we’ll see the
GSP Demonstration
anyway.
Parameter
What It Does
d
Vertical Translation
• Up: d > 0; down: d < 0
a
Vertical Scaling
• Stretch: |a| > 1; Shrink: 0 < |a|< 1; Reflect across x: a < 0
Amplitude: Half the vertical distance from minimum to
maximum points
b
Horizontal Scaling
• Stretch: 0 < |b| < 1; Shrink: |b| > 1;
• (Sine only) Reflect across y: b < 0
c
Horizontal Translation
•Left: x + c ; Right x − c
Recall that cosine is an even function and sine is
an odd function.
cos(t )  cos t
sin(t )   sin t
• This explains why a negative b-value does not
reflect the cosine curve across the y-axis.
• Notice that a negative b-value can also be
interpreted as a reflection across the x-axis for
a sine curve.
Which of the parameters affect the range of the
sine/cosine curve?
y  d  a sin  bx  c 
You will be able to find the period, amplitude, and
phase shift of sine or cosine curves
Determine the period and the 5 key points of
each function.
1. y = sin x
2. y = sin (6x)
The period of sin x and cos x is 2π.
…
y  d  a sin  bx  c 
y  d  a cos  bx  c 
For the general sinusoid:
• Period: 2π/b
• Interval for 5 key points: period/4 or
(1/4)∙period
Determine the period and the 5 key points of
each function.
1. y = sin (3x)
2. y = sin (x/3)
3. y = sin (2πx)
4. y = cos (5πx/3)
Determine the horizontal shift of each function.
1. y = cos (x – π)
2. y = cos (2x + π)
y  d  a sin  bx  c 
y  d  a cos  bx  c 
Whereas c represents a horizontal shift, the
graph does not always shift by exactly c units.
 
c 
y  d  a sin b  x   
b 
 
 
c 
y  d  a cos b  x   
b 
 
When you factor out b, you can see that the
horizontal translation is actually c/b. This is
called the phase shift.
A person’s blood pressure changes in a rhythmic
fashion, with a periodicity set by the beating
of the heart. Suppose that a person’s blood
pressure at time t seconds is given by:
P(t )  100  25sin(160 t )
1. Determine the rate at which the person’s
heart is beating.
2. What are the maximum and minimum blood
pressure readings?
You will be able to graph sine and
cosine functions as transformations
on a parent function
Graph each of the following.
1. y = 3 sin x
2. y = sin (3x)
3. y = cos (x – π/2)
4. y = cos (2x – π/2)
5. y = 2 + .5 sin (3x + π)
6. y = cos (2πx)
• Sinusoidal: Looks like a sine curve
• Oscillate: To move back and forth between
two positions
• Wavelength: Same as period
Objectives:
1. To graph sine and
cosine functions as
transformations on a
parent function
2. To find the period,
amplitude, and
phase shift of sine or
cosine curves
•
•
•
•
•
•
•
Assignment:
P. 328: 1-14 S
P. 328: 15-26 S
P. 329: 27-34 S
P. 329: 35-56 S
P. 329: 63-70 S
P. 330: 73, 76, 77
P. 331: 81-83, 86