Graphing Trigonometric Functions Worksheet
The basic trig function is
y = sin x . We know that this has an amplitude of 1, period of 2π
, and no phase shift. The
graph is below. Make sure you know how to graph this and what they key points are!
What we want to do is sketch variations of y = sin. This refers to transformations.
The equation is as follows:
=
y
A sin (ω x − φ ) . You will need to memorize this!
A is the amplitude of the function. It tells us what the highest and lowest y-value should be. Note, if A is negative, that
means that the graph is reflected across the x-axis.
ω
is the value that tells how the period has changed. The period can be found by T
=
2π
ω
, where T is the period.
Remember, the period refers to the length of one complete wave.
φ
is the value that tells how the phase shift has changed. The phase shift is found by
φ
. If this value is positive, the
ω
phase shift is to the right. If this value is negative, the phase shift is to the left.
EXAMPLE
Find the amplitude, period, and phase shift
of y
=
First identify the three variables. A = 2,
2sin ( 2 x − π ) .
ω = 2 , and φ = π . Note that the value of φ
Next, calculate the period and the phase shift. The period is given by
of one complete wave is
π . Note that this is a horizontal shrink.
Last, find the phase shift. The phase shift can be found by
graph is shifted to the right!
T=
2π
ω
here is positive!
T
. Here,=
2π
= π . Hence the length
2
π
φ
. Here, the phase shift is
. Note that this means the
2
ω
To sketch the graph, here is what I recommend:
=
First, determine where the starting point is. This can be found using the phase shift. The graph
of y
has a phase shift of
π
2
, so instead of starting at (0,0), the graph will begin at
2sin ( 2 x − π )
π 
 ,0  .
2 
Second, draw one complete wave. The graph of sin should start at the x-axis and then end at the x-axis.
Finally, fill in the values of the amplitude and the critical points – where the graph crosses the x-axis and where the
graph has local max and min. See the graph below.
Note that you really only need to draw one complete period, so the graph can start at
π
2
and then end at
PRACTICE
1) Find the amplitude, period and phase shift, and sketch one complete period
of y
=
π

3sin  3 x + 
2

2) Find the amplitude, period and phase shift, and sketch one complete period
of y
=
π
1
sin  x + 
3
2
Now let’s sketch
y = cos x . Use the Unit Circle to get the (x,y) ordered pairs. The sketch is below.
3π
2
.
Note that this starts at (0,1) since cos 0 = 1!
The sketch
of y
=
A cos (ω x − φ )
is done the same way as with sin.
PRACTICE
3) Find the amplitude, period and phase shift, and sketch one complete period
of y
=
1

2cos  x − π 
2

4) Find the amplitude, period and phase shift, and sketch one complete period
of y
=
π
2
2cos  x + 
3
3
Now let’s sketch the graph of the reciprocal functions.
Try graphing
y = csc x . Remember, csc x is the reciprocal of sin x. Hence, if we know what the graph of sin x looks like,
we can sketch the graph of csc x by taking the reciprocal of the respective y-values!
x
0
π/6
π/4
π/3
π/2
sin(x)
0
½
√2/2
√3/2
1
csc (x)
DNE
2
√2
2√3/3
1
The graph in red is the graph of y = sin x. The graph in blue is the graph of y =csc x. Do you see any relationship between
the two?
We see that where sin x = 0, that is where csc x has vertical asymptotes! Also, the range of sin x is [-1,1] so the range of
csc x is everything EXCEPT [-1,1]!
So to sketch the graph of y = csc x, we first sketch the graph of y = sin x and use that graph as a guide. The same with the
graph of y = sec x. We would draw the corresponding y = cos x graph first.
EXAMPLE
π

2csc  2 x − 
3

Sketch the graph
of y
=
Note that we can sketch the graph of the sin function first, and then use that as a guide to sketch the graph of csc.
We see that the amplitude is 2, the period is π, and the phase shift is 𝜋�6 to the right.
Note that this means we start the graph of sin x at 𝜋�6 , and since one complete period is π, the graph will essentially
‘end’ at
π
7π
+π =
6
6
.
Now we want to find the 3 values in between – the max and min values and where the graph crosses the x-axis. See the
graph below.
Remember, this is the graph
of y
=
π
π


2sin  2 x −  . To get the sketch
of y 2csc  2 x −  , we need to draw
=
3
3


vertical asymptotes at the x-intercepts, and use the graph of sin as a guide. The final answer is below.
PRACTICE
1
3
1 
4 
5) Sketch one complete period of the graph
of y sec  x − π  . Make sure you label the vertical asymptotes.
=
1 
 3π
x + π  . Make sure you label the vertical asymptotes.
3 
 2
6) Sketch one complete period of the graph
of y csc 
=