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Trigonometric Trigonometric GraphsGraphs
Written by B. Willox (Bridge of Don Academy) for Fourth Year – Credit Level
Click to continue.
You are already familiar with the basic graph of y = sin xo.
There are some important points to remember.
360o
The curve has a period of
It has a maximum value
of 1 at 90o.1
90o
It has a minimum value of –1 at 270o.
-1
270o
It passes through the origin. O
It crosses the x-axis at
180o
Click to continue.
y = sin xo
x
y
Let us compare the graph of y = sin xo to the family of graphs of the form y = a sin bxo + c
where a, b and c are constants.
We will begin by looking at graphs of the form y = a sin xo.
Click to continue.
For example: y = 2 sin xo,
y = 3.7 sin xo or y = ½ sin xo.
Click to continue.
y = sin xoO 180o
360o
1
-3
-2
3
2
-1
Here is the graph of y = sin xo.
Click once to see the graph of y = 2 sin xo.
y = 2 sin xo Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of 2 (twice that of the
normal graph).
It has a minimum of –2.
It has a period of
360o.
x
y
Click to continue.
O 180o360o
1
-3
-2
3
2
-1
Here is the graph of y = sin xo.
Click once to see the graph of
y = -3 sin xo.
y = -3 sin xoNotice the
following points on the upside-down curve.
It passes through
the origin.
It has a minimum of -3 (negative three times
that of the normal graph).
It has a maximum of 3.
It has a period of 360o.
x
y
y = sin xo
Click to continue.
O 180o360o
1
-3
-2
3
2
-1
Here is the graph of y = sin xo.
Click once to see the graph of
y = 2½ sin xo.
y = 2½ sin xo Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of 2½ (two and a half
times that of the normal graph).
It has a minimum of –2½.
It has a period of 360o.
x
y
y = sin xo
Click to continue.
O 180o360o
1
-3
-2
3
2
-1
Here is the graph of y = sin xo.
Click once to see the graph of
y = ½ sin xo.
y = ½ sin xo
Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of ½ (half of the
normal graph).
It has a minimum of –½.
It has a period of 360o.
x
y
y = sin xo
Click to continue.
O 180o360o
1
-a
a
-1
Here is the graph of y = sin xo.
Click once to see the graph of y = a sin xo.
y = a sin xo Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of a (a times that of the
normal graph).
It has a minimum of –a.
It has a period of 360o.
x
y
y = sin xo
It still passes through the origin
The period is unaffected.
The height is now “a”.
The height is now “a”.
y = sin 15xo
O 180o360o
5
-15
-10
15
10
-5
x
yThis is the graph of which function?
y = 15 sin xo
y = sin xo + 15
WRONG!
Try again.
Well Done!
Click to continue
Which of these diagrams shows the graph of y = 7 sin xo?
x
y
O 180o 360o
7
-7
14
-14
x
y
O 360o 720o
7
-7
x
y
O 360o 720o
7
-7
x
y
O 180o 360o
-3.5
3.5
540o180o
-7
-3.5
3.5
7
WRONG! The height (or altitude) is too big.
Try again.
Well Done!
Click to continue
WRONG! The period is too long.
Try again.
WRONG! The period is too short.
Try again.
WRONG! The period is too short.
Try again.
Click to continue.
For y = a sin xo only the height is affected.
The graph will now have an altitude of 1 a.
This is also true for y = a cos xo and y = a tan xo.
90o 180o 270o 360oO x
y
-1
1
y = cos xo
90o 180o 270o 360oO x
y
-1
1
45o
y = tan xo
Here are the graphs of y = cos xo and y = tan xo.
Click to continue.
Here are some examples of the graphs of y = a cos xo.
90o 180o 270oO
Click for
y = 2 cos xo
Click for y = ¾ cos xo
Click for
y = - cos xo
y = cos xo
360o x
y
1
-1
90o 180o 270o 360oO x
y
-1
1
45o
Click to continue.
Here are some examples of the graphs of y = a tan xo.
450o
y = tan xy = tan xoo
Click for
y = 2tan xo
3
2
4
-2
-3
-4
Click for
y = -3tan xo
Notice this
point
Notice this
point
Notice this
point
We will now look at graphs of the form y = sin bxo.
Click to continue.
For example: y = sin 2xo,
y = sin 3xo or y = sin ½xo.
You are already familiar with the basic graph of y = sin xo.
There are some important points to remember.
360o
1
90o
-1
270oO 180o
Click to continue.
y = sin xo
x
y
Click to continue.
y = sin xo
O 180o360o
1
-1
Here is the graph of y = sin xo.
Click once to see the graph of y = sin 2xo.
y = sin 2xo
Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of 1 (the same as a normal graph).
It has a minimum of –1.
It has a period of 360o ÷ 2 = 180o.
x
y
Click to continue.
y = sin xo
O 180o360o
1
-1
Here is the graph of y = sin xo.
Click once to see the graph of y = sin 3xo.
y = sin 3xo
Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of 1 (the same as a normal graph).
It has a minimum of –1.
x
y
It has a period of 360o ÷ 3 = 120o.
Click to continue.
y = sin xo
O 180o 360o
1
-1
Here is the graph of y = sin xo.
Click once to see the graph of y = sin ½xo.
y = sin ½ xo
Notice the following points
on the curve.
It passes through the
origin.
It has a maximum of 1 (the same as a normal graph).
It has a minimum of –1.
It has a period of 360o ÷ ½ = 720o.
x
y
540o 720o
Click to continue.
O
Period is (360o ÷ b)
1
-1
Here is the graph of y = sin bxo.
y = sin bxo
x
y
It still passes through the
origin.
The altitude (or height) is
unaffected.
The period is 360o b.
The period is 360o ÷ b.
y = 4 sin xo
O 180o
-1
1
x
yThis is the graph of which function?
y = sin 2xo
y = sin 4xo
90o45o 135o
WRONG! Remember, a normal SINE graph has a period
of 360o.
Try again.
Well Done!
Click to continue
Which of these diagrams shows the graph of y = sin 6xo?
x
y
O 180o 360o
1
-1
x
y
O 90o 180o
1
-1
x
y
O 60o 120o
1
-1
x
y
O 45o 90o
-0.5
0.5
90o30o
-1
-0.5
0.5
1
WRONG! Remember, the period of a normal SINE
graph is 360o.
Try again.
Well Done!
Click to continue
Click to continue.
For y = sin bxo only the period is affected.
The graph will now have a period of 360o b.
This is also true for y = cos bxo and y = tan bxo.
90o 180o 270o 360oO x
y
-1
1
y = cos xo
90o 180o 270o 360oO x
y
-1
1
45o
y = tan xo
Here are the graphs of y = cos xo and y = tan xo.
Click to continue.
Here are some examples of the graphs of y = cos bxo.
90o 180o 270oO
Click for
y = cos 2xo period = 360o ÷ 2 = 180o
Click for
y = cos 2/3 xo
period = 360o ÷ 2/3 = 540o
Click for
y = cos ½xo
period = 360o ÷ ½ = 720o
y = cos xo
360o
y
450o 540o 630o
1
-1
y720o
90o 180o-45o-90o O x
y
-1
1
45o
Click to continue.
Here are some examples of the graphs of y = tan bxo.
y = tan xo
3
2
4
-2
-3
-4
Notice this
point
Notice this
point
Click to see
y = tan 2xo period = 180o ÷ 2 = 90o
and
45o ÷ 2 = 22.5o
y = tan 2xo
Click to see
y = tan ½xo period = 180o ÷ ½ = 360o
and
45o ÷ ½ = 90o
Notice this
point
y = tan ½xo
We will now look at graphs of the form y = sin xo + c.
Click to continue.
For example: y = sin xo + 2,
y = sin xo + 3 or y = sin xo – 1.
You are already familiar with the basic graph of y = sin xo.
There are some important points to remember.
360o
1
90o
-1
270oO 180o
Click to continue.
y = sin xo
x
y
Click to continue.
y = sin xoO 180o
360o
1
-1
Here is the graph of y = sin xo.
Click once to see the graph of y = sin xo + 1.
y = sin xo + 1 Notice the following
points on the curve.
It passes through the origin + 1 = (0, 1).
It has a maximum of 1 + 1 = 2.
It has a minimum of –1 + 1 = 0.
It has a period of 360o.
x
y
2
-2
3
-3
The whole graph has
been moved up one unit.
The whole graph has
been moved up one unit.
Click to continue.
Here is the graph of y = cos xo.
90o 180o 270oO
y = cos xo
360o x
y
1
-1
Click once to see the graph of y = cos xo – 1.
y = cos xo – 1
The whole graph has been moved down one unit.The whole graph has been moved down one unit.
90o 180o 270o 360oO x
y
-1
1
45o
Click to continue.
Here is the graph of y = tan xo.
450o
y = tan xo3
2
4
-2
Notice this
point
Notice this
point
Click once to see the graph of y = tan xo + 2.
The whole graph has been moved up two units.The whole graph has been moved up two units.
y = tan xo + 2
y = -3 sin xo
O 720o
-1
1
x
yThis is the graph of which function?
y = sin xo + 2
y = sin xo – 2360o180o 540o
2
3
-2
-3
WRONG! Remember, a normal SINE graph has a height
of 2 (from –1 to 1).
Try again.
Well Done!
Click to continue
Which of these diagrams shows the graph of y = cos xo + 2?
x
y
O 180o 360o
2
-2
4
-4
x
y
O 360o
2
-2
x
y
O 360o
6
-2
2
4
540o180o
180o 540o
y
O 180o 360o
-1
2
3
1
x
WRONG! This is the graph of y = sin xo + 2.
Try again.
WRONG! This is the graph of y = cos xo + 1.
Try again.
Well Done!
Click to continue
WRONG! This is the graph of y = 2 cos xo + 3.
Try again.
We will now look at graphs of the form y = a sin bxo + c,
y = a cos bxo + c and
y = a tan bxo + c.
Click to continue.
For example: y = 2 sin 3xo – 1,
y = ½ cos 4xo + 3 or y = ¾ tan ¼xo – 12.
Let us look at the graph of y = 2 sin 3xo – 1.
Begin by considering the simple curve of y = sin xo.
180o 540o360o
x
y
O
Now, think on the graph of y = 2 sin xo: the 2 will double the height.
The graph of y = 2 sin 3xo: the 3 makes the period as long (360o ÷ 3 = 120o)
1
2
-1
-2
-3
120o
Finally, y = 2 sin 3xo – 1, where the –1 moves the whole graph down one unit.
y = 2 sin 3xo – 1
Click to continue.
Look at this graph. What function does it show?
180o
y
O
1
2
-1
-2
-3
x360o90o 270o
2. Next, look at the height.
Maximum of 0.5
Minimum of –2.5
Therefore, the height is 3 units.Normally, a COSINE graph has a height of 2. Therefore the height has been multiplied by 3 ÷ 2 = 1.5
1. First, decide on the type.
3. Now, consider the period.
The first complete wave finishes here.
This means the period is 180o
so 360o ÷ 180o = 2.
4. Finally, find out how much it has been moved down (or up).
This is the middle of the wave and it has been moved 1 unit down from the x-axis.
It must be a COSINE graph because the first bump is on the y-axis.
a = 1.5 b = 2 c = - 1
Therefore, we get –1.
y = 1.5 cos 2xo - 1 y = 1.5 cos 2xo - 1
Click to continue.
Which of these graphs shows the function y = 2 sin 3xo + 1?
x
y
O 180o 360o
1
-1
x
y
O 360o
2
-2
x
y
O 360o
6
-2
2
4
1080o720o
180o 540o
y
O 120o 240o
-1
2
3
1
x
2
-3-2
WRONG! This is the graph of y = 2 sin 2xo + 1.
Try again.
WRONG! This is the graph of y = cos xo + 1.
Try again.
WRONG! This is the graph of y = 4 cos ½xo + 1.
Try again.
Well Done!
Click to continue
Trigonometric Trigonometric Graphs Graphs
Presentation is Presentation is complete.complete.
Then go to http://www.univie.ac.at/future.media/moe/galerie/trig/trig.html and select The graphs of sin, cos and tan from the bottom of the screen. Then scroll down and click on the red boxes for “Recognize functions 3”
and/or “Recognize graphs 3”.
Want to try some harder questions?