Graphing Trigonometric Functions

A periodic function repeats a pattern of y-values (outputs) at regular intervals.

One complete pattern is called a cycle.

The amplitude of y = a sin x and y = a cos x represents half the diasance between the maximum and minimum values of the function and is given by...     Amplitude =  | a |

The sin wave:

The zeros of y = sin x are at the multiples of π.  And it is there that the graph crosses the x-axis, because there sin x = 0.  But what is the maximum value of the graph, and what is its minimum value?

 

Sin x has a maximum value of 1 at π/2, and a minimum value of – 1 at 3 π/2 – and all angels coterminal with them.

Heres the graph of a y= sin(x) 

Properties of the sin function:

Suppose:  y = a sin b θ,   with a ≠0,    b>0,    θ in radians

-          | a | is the amplitude of the function (max/min)

-          b is the number of cycles in the interval from 0 to 2 π

-          2 π/b is the period of the function

-          Graphing a sin function à cut period in half twice

y = sin(x)  (odd function)

-           

- Period = 2 π/b (maxà0minà0àmax) 

-     1 cycle (one entire curve) in 2 π



The cosine wave:

Properties of the cosine function:

Suppose:  y = a cos b θ, with a≠

0, b>0, θ in radians

-        | a | is the amplitude of the function (max/min)

-          b is the number of cycles in the interval from 0 to 2 π

-          2 π/b is the period of the function

y = cos(x)





The graphs of trigonometric functions can be transformed the same way as any other graph...

Consider the function y = a sin [b(x –c)] + d, where a, b, c, and d are constants.  Explain how the value of each of these constants affects the graph of the parent function y = sin x.

 a) Vertically stretches of compresses (can also reflect the x axis if negative)

b) Horizontally stretch of compress

c) moves left or right (phase shift)

d) Shifts up or down (mid-line shift)