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Lesson Objectives Able to plot the different Trigonometric Graphs Graph of Sine Function (y = f(x) = sinx) Graph of Cosine Function (y = f(x) = cosx) Define the Maximum and Minimum value in a graph Generalized Trigonometric Functions Graphs of y = sinbx Graphs of y = sin(bx + c) Could find the Period of Trigonometric Functions Could find the Amplitude of Trigonometric Functions Variations in the Trigonometric Functions
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Presents:
GRAPHS OF SINE AND COSINE FUNCTIONS
credit: Shawna Haider
GRAPHS OF
We are interested in the graph of y = f(x) = sin x
Start with a "t" chart and let's choose values from our unit
circle and find the sine values.
x y = sin x
6
0 0
2
1
2
1
6
5
2
1We are dealing with x's and y's on the unit circle
to find values. These are completely different
from the x's and y's used here for our function.
x
y
1
- 1
plot these points
y = f(x) = sin xchoose more values
x y = sin x
6
7
0
2
1
2
31
6
11
2
1
If we continue picking values for x we will start
to repeat since this is periodic.
x
y
1
- 1
plot these points
2 0
join the points
6
2
Here is the graph y = f(x) = sin x showing
from -2 to 6. Notice it repeats with a
period of 2.
It has a maximum of 1 and a minimum of -1 (remember
that is the range of the sine function)
2 22 2
What are the x intercepts? Where does sin x = 0?
0 2 3 423
…-3, -2, -, 0, , 2, 3, 4, . . .
Where is the function maximum? Where does sin x = 1?
2
2
5
2
3
2
7
2
5,
2,
2
3,
2
7
Where is the function minimum? Where does sin x = -1?
0 2 3 423
2
2
5
2
3
2
7
2
7,
2
3,
2,
2
5
2
5
2
2
3
2
7
Thinking about transformations that you learned
and knowing what y = sin x looks like, what do
you suppose y = sin x + 2 looks like?
The function value
(or y value) is just
moved up 2.
y = sin x
y = 2 + sin x This is often written
with terms traded
places so as not to
confuse the 2 with
part of sine function
Thinking about transformations that you've
learned and knowing what y = sin x looks like,
what do you suppose y = sin x - 1 looks like?
The function value
(or y value) is just
moved down 1.
y = sin x
y = - 1 + sin x
Thinking about transformations that you learned
and knowing what y = sin x looks like, what do
you suppose y = sin (x + /2) looks like?
This is a horizontal
shift by - /2
y = sin x
y = sin (x + /2)
Thinking about transformations that you learned
and knowing what y = sin x looks like, what do
you suppose y = - sin (x )+1 looks like?
This is a reflection about
the x axis (shown in
green) and then a
vertical shift up one.
y = sin x
y = - sin x
y = 1 - sin (x )
What would the graph of y = f(x) = cos x look like?
We could do a "t" chart and let's choose values from our
unit circle and find the cosine values.
x y = cos x
3
0 1
2
1
2
0
3
2
2
1
We could have used the same values as we did
for sine but picked ones that gave us easy
values to plot.
x
y
1
- 1
plot these points
6
y = f(x) = cos x Choose more values.
x y = cos x
3
4
1
2
1
2
30
3
5
2
1
cosine will then repeat as you go another loop
around the unit circle
x
y
1
- 1
plot these points
6
2 1
Here is the graph y = f(x) = cos x showing
from -2 to 6. Notice it repeats with a
period of 2.
It has a maximum of 1 and a minimum of -1 (remember
that is the range of the cosine function)
2 22 2
Recall that an even function (which the cosine is)
is symmetric with respect to the y axis as can be
seen here
What are the x intercepts? Where does cos x = 0?
2
2
3
2
5
2
2
3
…-4, -2, , 0, 2, 4, . . .
Where is the function maximum? Where does cos x = 1?
0 22
2
5,
2
3,
2,
2,
2
3
2
2
3
2
542
2
3
…-3, -, , 3, . . .
Where is the function minimum?
0 22
Where does cos x = -1?
33
You could graph transformations of the cosine function the
same way you've learned for other functions.
Let's try y = 3 - cos (x - /4)
reflects over x axis
moves up 3 moves right /4
y = cos x y = - cos x
y = 3 - cos x y = 3 - cos (x - /4)
What would happen if we multiply the function by a
constant?
y = 2 sin x
All function values would be twice as high
y = 2 sin x
y = sin x
The highest the graph goes (without a vertical shift) is
called the amplitude.
amplitude
of this
graph is 2
amplitude is here
For y = A cos x and y = A sin x, A is the amplitude.
y = 4 cos x y = -3 sin x
What is the amplitude for the following?
amplitude is 4 amplitude is 3
The last thing we want to see is what happens if we put
a coefficient on the x.
y = sin 2x
y = sin 2x
y = sin x
It makes the graph "cycle" twice as fast. It does one
complete cycle in half the time so the period becomes .
What do you think will happen to the graph if we put a
fraction in front?
y = sin 1/2 x
y = sin x
The period for one complete cycle is twice as long or 4
xy2
1sin
So if we look at y = sin x the affects the
period.
The period T =
2This will
be true for
cosine as
well.
What is the period of y = cos 4x?
24
2 T
This means
the graph
will "cycle"
every /2 or
4 times as
often y = cos 4x
y = cos x
tAy cos tAy sin
absolute value of this
is the amplitude
Period is 2 divided by this
Sample Problem
• Which of the following
equations best describes
the graph shown?
(A) y = 3sin(2x) - 1
(B) y = 2sin(4x)
(C) y = 2sin(2x) - 1
(D) y = 4sin(2x) - 1
(E) y = 3sin(4x)
2 1 1 2
5
4
3
2
1
1
2
3
4
5
Sample Problem
• Find the baseline between
the high and low points.
– Graph is translated -1
vertically.
• Find height of each peak.
– Amplitude is 3
• Count number of waves in
2
– Frequency is 2
2 1 1 2
5
4
3
2
1
1
2
3
4
5
y = 3sin(2x) - 1
• Which of the following equations best describes the graph?
– (A) y = 3cos(5x) + 4
– (B) y = 3cos(4x) + 5
– (C) y = 4cos(3x) + 5
– (D) y = 5cos(3x) + 4
– (E) y = 5sin(4x) + 3
Sample Problem
2 1 1 2
8
6
4
2
• Find the baseline
– Vertical translation + 4
• Find the height of
peak
– Amplitude = 5
• Number of waves in
2
– Frequency =3
Sample Problem
2 1 1 2
8
6
4
2
y = 5 cos(3x) + 4
