## Trigonometric Graphs The graphs of the functions of the three trigonometic ratios, y = sin x, y = cos x and y = tan x are shown below.

These graphs are called periodic functions because the pattern repeats itself over and over. The frequency with which it does this is called the period of the function.

The amplitude of the graph is the distance between the x-axis and the highest point on the graph.

### y=sin x

The period of the graph is 2π or 360°.

The amplitude of the graph is 1.

There are no asymptotes.

The domain is R the set of real numbers.

The range is
-1≤ y ≤ 1 Click Here to see the graph of y = sinx being formed.

### y=cos x

The period of the graph is 2π or 360°.

The amplitude of the graph is 1.

There are no asymptotes.

The domain is R the set of real numbers.

The range is
-1≤ y ≤ 1 Click Here to see the graph of y = cosx being formed.

### y=tan x

The period of the graph is π radians or 180°.

There are asymptotes every π radians or 180°.

The domain is R the set of real numbers except where the asymptotes occur.

The range is the real numbers R Click Here to see the graph of y = tanx being formed.

### Transformations of the Basic Trigonometric Functions

More complicated trigonometric functions of the form y = Asin(Bx − C) + D and y = Acos(Bx − C) + D can be sketched by transforming the basic graphs shown above. Each of the variables A, B, C and D change either the shape or position of the basic graphs.

 Type of function Feature changed Transformation required Examples y = A sinx y = A cos x the amplitude The basic graph is compressed or stretched along y-axis by scale factor A. The new ampitude is A y = 2sin x y = -3cos x y = sin Bx y = cos Bx the period The basic graph is compressed or stretched along x-axis by scale factor B. The new period is 360° ÷ B y = sin 2x y = cos 0.5x y = sin(x − C) y = cos(x − C) the position The basic graph is translated along the x-axis by C. For (x − 30°) the graph moves 30°to the right.For (x + 30°) the graph moves 30° to the left. y = sin(x − 45°) y = cos(x + π/2) y = sinx + D y = cos x + D the position The basic graph is translated along the y-axis by D. If D is positive the graph moves UPIf D is negative the graph moves DOWN y = sin x + 2 y = cos x − 1

Putting it all together...

 Sketch the graph of y = 3sin(2x - 60) + 1 3 Changes amplitude to 3 2 Changes period to 360 ÷ 2 = 180° 2x - 60 Moves graph right by 60 ÷ 2 = 30° +1 Moves graph up 1

The graph looks like this: It is unlikely that you will be asked to perform all four of these transformations for one graph!