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Provided by 1 How to Graph Trigonometric Functions
Created September 2013
How to Graph Trigonometric Functions
This handout includes instructions for graphing processes of basic, amplitude shifts,
horizontal shifts, and vertical shifts of trigonometric functions.
The Unit Circle and the Values of Sine and Cosine Functions
The unit circle is a circle with a radius that equals 1. The angle θ is formed from the φ
(phi) ray extending from the origin through a point p on the unit circle and the x - axis; see
diagram below. The value of sin θ equals the y-coordinate of the point p and the value of
cos θ equals the x- coordinate of the point p as shown in the diagram below.
p = (cos θ, sin θ)
θ
This unit circle below shows the measurements of angles in radians and degrees. Beginning
at 0π, follow the circle counter-clockwise. As angle θ increases to
π
2
radians or 90°, the value
of cosine (the x - coordinate) decreases because the point is approaching the y-axis.
Meanwhile, the value of sine (the y-coordinate) increases. When one counter-clockwise
revolution has been completed, the point has moved 360° or 2π.
π or 90°
0π or 0°
π or 180°
2π or 360°
3 𝜋
2
or 270°
Provided by 2 How to Graph Trigonometric Functions
Created September 2013
Graphing Sine and Cosine Functions y = sin x and y = cos x
There are two ways to prepare for graphing the basic sine and cosine functions in the form
y = sin x and y = cos x : evaluating the function and using the unit circle.
To evaluate the basic sine function, set up a table of values using the intervals 0π,
π
2
3π
2
, and 2π
for x and calculating the corresponding y value.
f(x) or y = sin x
f(x) or y x
0 0 π
π
0 π
3 π
0 2 π
To use the unit circle, the x-coordinates remain the same as within the list above. To find the
y- coordinate of the point to graph, first locate the point p on the unit circle that corresponds
to the angle θ given by the x-coordinate. Then, use the y-coordinate of the point p as the y
value of the point to graph.
To draw the graph of one period of sine or y = sin x , label the x-axis with the values 0π,
π
2
, π,
3π
2
and 2π. Then plot points for the value of f(x) or y from either the table or the unit circle.
Provided by 4 How to Graph Trigonometric Functions
Created September 2013
To use the unit circle, the x-coordinate remains the same as the list on the previous page. To
find the y-coordinate of the point to graph, first locate the point p on the unit circle that
corresponds to the angle θ given by the x - coordinate. Then, use the x-coordinate of the
point p as the y value of the point to graph.
To draw the graph of one period of cosine or y = cos x , label the x-axis with the values 0π,
𝜋
2
, π,
3 𝜋
2
, and 2π. Then plot points for the value of f(x) or y from either the table or the unit circle.
Add other points as required for the intermediate values between those above to obtain a
more complete graph, and draw a best fit line connecting the points. The graph below shows
one and a half periods.
0π
π
π
3π 2π
One period
5π
3π
y = cos x
0π
π
π
3π
2π
Provided by 5 How to Graph Trigonometric Functions
Created September 2013
Graphing the Tangent Function y = tan x
The tangent value at angle θ is equal to the sine value divided by the cosine value (
𝑆𝑖𝑛𝑒 𝑉𝑎𝑙𝑢𝑒
𝐶𝑜𝑠𝑖𝑛𝑒 𝑉𝑎𝑙𝑢𝑒
of the same angle θ. The value of tangent at 0π for the unit circle is
0
1
, which is equivalent to 0.
The value of tangent at
𝜋
2
is
1
0
. This yields a divide by 0 error or undefined (try this in your
calculator). Therefore, the tangent function is undefined at
𝜋
2
. This is illustrated by drawing an
asymptote (vertical dashed line) at
𝜋
2
. See the figure below.
The value of tangent at π is
0
1
, which results in 0. To determine how the tangent behaves between
0π and the asymptote, find the sine and cosine values of
π
4
, which is half way between 0π and
π
2
Looking at the handout Common Trigonometric Angle Measurements , the tangent of
π
4
is
√ 2
2
(sine)
divided by
√ 2
2
(cosine). Flipping the cosine value and multiplying gives
√ 2
2
×
2
√ 2
which simplifies to 1.
The value of tangent at
π
4
is therefore 1. These points have been added to the graph below.
0π
π
π
3π
2π
0π
π
π
π
3π
2π
Provided by 7 How to Graph Trigonometric Functions
Created September 2013
Amplitude Shifts of Trigonometric Functions
The basic graphs illustrate the trigonometric functions when the A value is 1. This A = 1 is
used as an amplitude value of 1. If the value A is not 1, then the absolute value of A value is
the new amplitude of the function. Any number
A
greater than 1 will vertically stretch the
graph (increase the amplitude) while a number
A
smaller than 1 will compress the graph
closer to the x - axis.
Example: Graph y = 3 sin x.
Solution : The graph of y = 3 sin x is the same as the graph of y = sin x except the
minimum and maximum of the graph has been increased to - 3 and 3 respectively
from - 1 and 1.
Horizontal Shifts of Trigonometric Functions
A horizontal shift is when the entire graph shifts left or right along the x - axis. This is shown
symbolically as y = sin(Bx – C). Note the minus sign in the formula. To find the phase shift (or
the amount the graph shifted) divide C by B (
C
). For instance, the phase shift of y = cos(2x – π)
can be found by dividing π (C) by 2 (B) , and the answer is
𝛑
𝟐
. Another example is the phase shift of
y = sin(-2x – π) which is – π (C) divided by - 2 (B) , and the result is
𝛑
𝟐
. Be careful when dealing
Amplitude
is now 3 up
π
π
Amplitude
is now 3 down
π
3π
2π
5π
3π
y = 3 sin x
Provided by 8 How to Graph Trigonometric Functions
Created September 2013
with the signs. A positive sign takes the place of the double negative signs in the form
y = sin(x + π). The C is negative because this example is also written as y = sin(x-(- π)) , which
produces the negative π phase shift (graphed below). It is important to remember a positive
phase shift means the graph is shifted right or in the positive direction. A negative phase shift
means the graph shifts to the left or in the negative direction.
Period Compression or Expansion of Trigonometric Functions
The value of B also influences the period, or length of one cycle, of trigonometric functions.
The period of the basic sine and cosine functions is 2π while the period of the basic tangent
function is π. The period equation for sine and cosine is: Period =
𝟐𝛑
|𝐁|
. For tangent, the
period equation is: Period =
𝛑
| 𝐁
|
. Period compression occurs if the absolute value of B is
greater than 1 ; this means the function oscillates more frequently. Period expansion occurs
if the absolute value of B is less than 1 ; this means the function oscillates more slowly.
The starting point of the graph is determined by the phase shift. To determine the key points
for the new period, divide the period into 4 equal parts and add this part to successive x values
beginning with the starting point.
Phase shift = - π
π
π
π
3π
2π
y = sin(x + π)
Provided by 10 How to Graph Trigonometric Functions
Created September 2013
Strategies, Summary, and Exercises
By using the following guidelines, it will make trigonometric functions easier to graph:
Recall the values of sine and cosine on the unit circle.
Identify the amplitude shift value A , in y = A sin x, y = A cos x, etc.
Identify the horizontal shift value
𝑪
𝑩
, in y = sin( B x – C ), y = cos( B x – C ), etc.
- The period compression or expansion of the graph is determined by dividing
the period of the basic function by the absolute value of B.
Identify the vertical shift value D , in y = sin (x) + D , y = cos (x) + D , etc.
The phase shift (if any) is the starting point to graphing the function. Divide the
period by 4 to determine the 5 points to graph on the x - axis.
- Use the 5 points to determine corresponding y-coordinate.
Exercises:
Find the amplitude shift, horizontal shift, period, and vertical shift of the following and graph
one period.
- y =
1
4
sin x + 1
y = 2 sin x
y =
1
2
cos(– x + π) – 1
y = tan x – 1
0π
π
π
3π
π 5π
3π
2π
Provided by 11 How to Graph Trigonometric Functions
Created September 2013
Solutions to Exercises
- y =
1
4
sin x + 1
First find the values of A, B, C, and D
A =
1
4
, B = 1, C = not written which means 0 , and D = 1
Amplitude shift is A, A =
1
4
Horizontal shift is
𝐶
𝐵
=
0
1
= 0
Period is
2 𝜋
| 1 |
= 2π
Vertical shift is D = 1
Graph of one period of y =
1
4
sin x + 1
- y = 2 sin x
First find the values of A, B, C, and D
A =2, B = 1, C = not written which means 0, and D = 0
Amplitude shift is A, A = 2
Horizontal shift is
𝐶
𝐵
=
𝟎
𝟏
= 0
Period is
𝟐𝝅
|𝟏|
= 2π
Vertical shift is D = 0
0π
π
π
3π
π
