E-learning

extended learning for chapter 11 (graphs)

Let’s recall first

Graph of y = sin

Graph of y = sin

Note: max value = 1 and min value = -1

Graph of y = sin

Graph of y = sin

The graph will repeats itself for every 360˚. The length of interval which the curve repeats is call the period.

Therefore, sine curve has a period of 360˚.

Graph of y = sin

Graph of y = sin 2

2

- 2

Graph of y = sin

In general, graph of y = sin a

a

- a

Graph of y = cos

Note: max value = 1 and min value = -1

Graph of y = cos 3

3

- 3

In general, graph of y = cos a

a

- a

Graph of y = cos

The graph will repeats itself for every 360˚.

Therefore, cosine curve has a period of 360˚.

Graph of y = tan

Note: The graph is not continuous. There are break at 90˚ and 270˚. The curve approach the line at 90˚ and 270˚. Such lines are called asymptotes.

45˚ 225˚

135˚ 315˚

In general, graph of y = tan a

a

- a

Note: The graph does not have max and min value.

y = tan

y = sin

y = cos

Summary

Identify the 3 types of graphs:

Points to consider when sketchingtrigonometrical functions:

• Easily determined points:a) maximum and minimum pointsb) points where the graph cuts the axes

• Period of the function

• Asymptotes (for tangent function)14

Let’s continue learning..

Example 1:Sketch y = 4sin x (given y = sin x ) for 0° x 360°

y = sin x

y = 4sin x

> x

x

xx

xx

x

xx x x

Comparing the 2 graphs, what happens to the max and min point of y = 4 sin x?

Example 2:Sketch y = 4 + sin x for 0° x 360°

y = sin x

y = 4 + sin x

> xx

x

x

x

x

x

x

x

x

x

Spot the difference between y = 4 sin x and y = 4 + sin x and write down the answer.

Example 3:Sketch y = - sin x for 0° x 360°

y = - sin x

> x

Reflection of y = sin x in x axis

x

x

x

x

x

x

x

xxx

y = sin x

How do we gety = - sin x graph from y = sin x?

Example 4:Sketch y = 4 - sin x for 0° x 360°

> x

y = 4 + (- sin x)

1. Reflection of y = sin x in x axis

2. Translation of y = -sin x by 4 units along y axis

x

x

x

xx

y = - sin x

y = sin x

Example 5:Sketch y = |sin x| for 0° x 360°

y = |sin x|

> x

y = sin x

Example 6:Sketch y = -|sin x| for 0° x 360°

> x

y = -|sin x|

1. Reflection of y = |sin x| in x axis

y = |sin x|

Sketch y = -5cos x for 0° x 360°

y = cos x y = 5cos x y = -5cos x

5

-5

Reflection about x axis

x

x

x

x

x

x

Example 7

Sketch y = 3 + tan x for 0° x 360°

y = tan x y = 3 + tan x

x

x

x

x

xx

x

x

Example 8

Sketch y = 2 – sin x, for values of x between 0° x 360°

y = sin x y = - sin x y = 2 – sin x

Example 9

Sketch y = 1 – 3cos x for values of x between 0° x 360°

y = 3 cos x y = - 3 cos x y = 1 – 3 cos x

y = 3 cos x

y = cos x

Example 10

Sketch y = |3cos x| for values of x between 0° x 360°

y = 3 cos x y = |3 cos x|

Example 11

Sketch y = |3 sin x| - 2 for values of x between 0° x 360°

y = 3 sin x y = |3 sin x| y = y = |3 sin x| - 2

y = 3 sin x

y = sin x

Example 12

Sketch y = 2cos x -1 and y = -2|sin x| for values of x between0 x 360.

Hence find the no. of solutions 2cos x -1 = -2|sin x| in the interval.

Solution:

Answer:

No of solutions = 2

x x

y = 2cos x -1

y = -2|sin x|

Example 13

x x

x x

Sketch y = |tan x| and y = 1 - sin x for values of x between0 x 360.

Hence find the no. of solutions |tan x| = 1 - sin x| in the interval.

Solution:

Answer:

No of solutions = 4

y = |tan x|

y = 1 - sin x

Example 14