extended learning for chapter 11 (graphs) E-learning extended learning for chapter 11 (graphs)

Let’s recall first

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

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 Graph of y = sin  2 - 2

In general, graph of y = sin  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

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

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

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

Let’s continue learning..

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?

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

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

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

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

Example 6: Sketch y = -|sin x| for 0°  x  360° y = |sin x| > x Reflection of y = |sin x| in x axis

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

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

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

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

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

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

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|

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