Trigonometric Trigonometric GraphsGraphs

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

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

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

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

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

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

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