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E-learning extended learning for chapter 11 (graphs)

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Lets recall first

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

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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˚.

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Graph of y = sin In general, graph of y = sin a a - a

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

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In general, graph of y = cos a a - 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 a - a Note: The graph does not have max and min value.

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

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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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Lets continue learning..

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

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

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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 x x y = - sin x y = sin x

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

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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|

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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

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

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

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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

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

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

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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 x x y = 2cos x -1 y = -2|sin x| Example 13

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x x x x 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| y = 1 - sin x Example 14

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