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.
plot these points
y
x
y = sin x
0

6

2
5
6
0
1
2
1
1
2
1
x
-1
We 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.
choose more values
x

7
6
3
2
y = sin x
plot these points
0
join the points
1

2
y
1
1
11
1

6
2
2
y = f(x) = sin x
0
-1

6

2 x
If we continue picking values for x we will start
to repeat since this is periodic.
Here is the graph y = f(x) = sin x showing
from -2 to 6. Notice it repeats with a
period of 2.
2
2
2
2
It has a maximum of 1 and a minimum of -1 (remember
that is the range of the sine function)
What are the x intercepts?
Where does sin x = 0?
…-3, -2, -, 0, , 2, 3, 4, . . .

7
2
 3

3
2
 2  

5
2
2
0

2
3
4
Where is the function maximum? Where does sin x = 1?
7
3  5

,
, ,

2
2 2 2
Where is the function minimum? Where does sin x = -1?
5
 3 7

, ,
,

2
2 2
2

7
2


3
2
2
 3  2  

5
2
5
2


2
0

2
3
2
3
7
2
4
Thinking about transformations that you learned
and knowing what y = sin x looks like, what do
you suppose y = sin x + 2 looks like?
y = 2 + sin x This is often written
with terms traded
places so as not to
confuse the 2 with
part of sine function
The function value
(or y value) is just
moved up 2.
y = sin x
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?
y = sin x
The function value
(or y value) is just
moved down 1.
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?
y = sin x
This is a horizontal
shift by - /2
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?
y = 1 - sin (x )
This is a reflection about
the x axis (shown in
green) and then a
vertical shift up one.
y = - sin x
y = 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.
plot these points
x
y = cos x
0

3

2
2
3
1
1
2
0
1

2
y
1
-1

6
x
We could have used the same values as we did
for sine but picked ones that gave us easy
values to plot.
y = f(x) = cos x
Choose more values.
x
y = cos x

4
1
1

2
3
plot these points
y
1
3
2
0
5
3
1
2
-1
2
1
cosine will then repeat as you go another loop
around the unit circle

6
x
Here is the graph y = f(x) = cos x showing
from -2 to 6. Notice it repeats with a
period of 2.
2
2
2
2
It has a maximum of 1 and a minimum of -1 (remember
that is the range of the cosine function)
Recall that an even function (which the cosine is)
is symmetric with respect to the y axis as can be
seen here
Where does cos x = 0?
What are the x intercepts?
3
  3 5

, , ,
,

2
2 2 2 2
 2
3  

2 2
2
0

2
3
2
5
2
Where is the function maximum? Where does cos x = 1?
…-4, -2, , 0, 2, 4, . . .
Where is the function minimum? Where does cos x = -1?
…-3, -, , 3, . . .
 2
3  

2 2
 3

2
0

2

3
2
5
2
3
4
You could graph transformations of the cosine function the
same way you've learned for other functions.
moves up 3
moves right /4
Let's try y = 3 - cos (x - /4)
reflects over x axis
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?
All function values would be twice as high
y = 2 sin x
amplitude
of this
graph is 2
amplitude is here
y = 2 sin x
y = sin x
The highest the graph goes (without a vertical shift) is
called the amplitude.
For y = A cos x and y = A sin x, A  is the amplitude.
What is the amplitude for the following?
y = 4 cos x
y = -3 sin x
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?
1
y  sin x
2
y = sin 1/2 x
y = sin x
The period for one complete cycle is twice as long or 4
So if we look at y = sin x the  affects the
period.
This will
The period T =
What is the period of y = cos 4x?
2

2 
T

4
2
be true for
cosine as
well.
y = cos x
This means
the graph
will "cycle"
every /2 or
4 times as
often
y = cos 4x
absolute value of this
is the amplitude
y  A cos t
y  A sin t
Period is 2 divided by this
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au