In these lessons, we will learn
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.
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
Since B = 2, the period is
The general form for the equation of the sine trigonometric function is
y = A sin B(x + p)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.
where A is the amplitude, the period is calculated by the constant B, and p is the phase shift.
A similar general form can be obtained for the other trigonometric functions.
Find the amplitude, period and phase shift of
The amplitude is 2, the period is π and the phase shift is units to the left.
Periodic Functions Definition, Period, Phase Shift, Amplitude, Vertical Shift.
Transformation of sin and cos with amplitude and vertical shift.
Transformation of sin and cos with period and phase shift.
Examples of transforming basic sine and cosine functions.