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**extended learning for chapter 11 (graphs)**

E-learning extended learning for chapter 11 (graphs)

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Let’s recall first

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**Graph of y = sin Note: max value = 1 and min value = -1**

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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

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Graph of y = sin 2 Graph of y = sin 2 - 2

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**In general, graph of y = sin a**

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Graph of y = cos Note: max value = 1 and min value = -1

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Graph of y = cos 3 3 - 3

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**In general, graph of y = cos a**

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**Graph of y = cos The graph will repeats itself for every 360˚.**

Therefore, cosine curve has a period of 360˚.

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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˚

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**In general, graph of y = tan a**

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

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Summary Identify the 3 types of graphs: y = sin y = cos y = tan

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**Points to consider when sketching trigonometrical functions:**

Easily determined points: a) maximum and minimum points b) points where the graph cuts the axes Period of the function Asymptotes (for tangent function) 14

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**Let’s continue learning..**

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

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

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Example 2: Sketch y = 4 + sin x for 0° x 360° Spot the difference between y = 4 sin x and y = 4 + sin x and write down the answer. y = sin x y = 4 + sin x > x x

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**Example 3: Sketch y = - sin x for 0° x 360° How do we get**

y = - sin x graph from y = sin x? x x y = - sin x x x x x x x > x y = sin x x x Reflection of y = sin x in x axis

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

Reflection of y = sin x in x axis Translation of y = -sin x by 4 units along y axis

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

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

Reflection of y = |sin x| in x axis

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**Example 7 Sketch y = -5cos x for 0° x 360° x 5 x x -5 x x x**

Reflection about x axis 5 -5 x x x x x x y = cos x y = 5cos x y = -5cos x

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**Example 8 Sketch y = 3 + tan x for 0° x 360° x x x x x x x x**

y = tan x y = 3 + tan x

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**Example 9 Sketch y = 2 – sin x, for values of x between 0° x 360°**

y = sin x y = - sin x y = 2 – sin x

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**Example 10 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

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**Example 11 Sketch y = |3cos x| for values of x between 0° x 360°**

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

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Example 12 Sketch y = |3 sin x| - 2 for values of x between 0° x 360° y = 3 sin x y = sin x y = 3 sin x y = |3 sin x| y = y = |3 sin x| - 2

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Example 13 Sketch y = 2cos x -1 and y = -2|sin x| for values of x between 0 x 360. Hence find the no. of solutions 2cos x -1 = -2|sin x| in the interval. Solution: Answer: No of solutions = 2 y = 2cos x -1 x x y = -2|sin x|

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Example 14 Sketch y = |tan x| and y = 1 - sin x for values of x between 0 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| x x y = 1 - sin x x x

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