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Transformation of Trigonometric Graphs

In this lesson, we will look at how trigonometric graphs can be transformed.

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function is the maximum displacement on the graph of that function.

In the case of sin and cos functions, this value is the leading coefficient of the function.
If y = A sin x, then the amplitude is .

In the case of tan, cot, sec, and csc, the amplitude would be infinitely large regardless of the value of A. However, for a limited domain, the value of A would determine the maximum height of these functions.

Period of Trigonometric Function

The period of a function is the displacement of x at which the graph of the function begins to repeat.

Consider y = sin x

The value x = 2π is the point at which the graph begins to repeat that of the first quadrant.
The coefficient of x is the constant that determine the period.

The general form is

y = A sin Bx

where is the amplitude and B determines the period.

For the functions sin, cos, sec and csc, the period is found by

Example:
Find the period of the graph y = sin 2x and sketch the graph of y = sin 2x for 0 ≤ 2x ≤ π.

Solution:
Since B = 2, the period is

Phase Shift of Trigonometric Functions

The general form for the equation of a trigonometric function is

y = A sin B(x + p)

where A is the amplitude, the period is calculated by the constant B, and p is the phase shift.
The graph y = sin x may be moved or shifted to the left or to the right. If p is positive, the shift is to the left; if p is negative the shift is to the right.