Graphs of Trig Functions

Graph of y = sin x y = sin x Max sin x = 1 when x = 90° Min sin x = -1 when x = 270°

Graph of y = 2sin x Max 2sin x = 2 when x = 90° Amplitude = 2 y = sin x Max 2sin x = 2 when x = 90° Min 2sin x = -2 when x = 270°

Graph of y = 5sin x Max 5sin x = 5 when x = 90° Amplitude = 5 y = sin x Max 5sin x = 5 when x = 90° Min 5sin x = -5 when x = 270°

The negative multiplier reflects the initial graph in the x-axis Graph of y = -sin x y = sin x Amplitude = 1 y = -sin x The negative multiplier reflects the initial graph in the x-axis

The negative multiplier reflects the initial graph in the x-axis Graph of y = -1·5sin x y = sin x Amplitude = 1·5 y = -1·5sin x The negative multiplier reflects the initial graph in the x-axis

Graph of y = cos x y = cos x Max cos x = 1 when x = 0° or 360° Min cos x = -1 when x = 180°

Graph of y = 3cos x and y = -3cos x

Write down the equations of the following graphs 2. 1. 4. 3.

6. 5. 8. 7.

Graph of y = sin x ± constant Adding or subtracting a constant to or from sin x moves the graph vertically by the amount of the constant

Graph of y = cos x ± constant

Graphs of Multiple Angles y = sin x y = sin 2x Max and Min values remain unchanged at +1 and -1 Two complete cycles between 0° and 360° Period = 180°

Graphs of Multiple Angles y = sin x y = sin 3x Max and Min values remain unchanged at +1 and -1 Three complete cycles between 0° and 360° Period = 120°

Graphs of Multiple Angles y = sin x y = sin 0·5x Max and Min values remain unchanged at +1 and -1 Half a complete cycle between 0° and 360° Period = 720°

Graphs of Multiple Angles y = cos x y = cos 4x Max and Min values remain unchanged at +1 and -1 Four complete cycles between 0° and 360° Period = 90°

Write down the equations and period of these graphs

b is the number of complete cycles between 0° and 360° Summary of Graphs so far y = a sin bx ± c or y = a cos bx ± c Where a, b and c are constants a is the amplitude, height above centre line of graph, if a is negative the graph is reflected in the x-axis b is the number of complete cycles between 0° and 360° c moves the graph vertically up or down from x-axis

Mixed examples – combined effects There are 2 complete cycles from 0° to 360° so y = 3sin 2x Amplitude is 3 so y = 3 sin ?x Curve rises from origin so is a sine graph

There is only one complete cycle from 0° to 360° so y = 6cos x + 4 Curve falls from y-axis so is a cosine graph Amplitude is 6, so y = 6 cos ?x + 4 Centre line half way between -2 and 10 = 4 So y = ? cos ?x + 4 There is only one complete cycle from 0° to 360° so y = 6cos x + 4

There are three complete cycles from 0° to 360° so y = 3cos 3x - 2 Curve falls from y-axis so is a cosine graph Amplitude is 3, so y = 3 cos ?x - 2 Centre line half way between -5 and 1 = -2 So y = ? cos ?x - 2

Find the equations of the following graphs