Determine the amplitude, the period, and the phase shift of...

Determine the amplitude, the period, and the phase shift of the function and, without a graphing calculator, sketch the graph of the function by hand. Then check the graph using
a graphing calculator.
$$y=\frac{1}{4} \cos x$$

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Key Concepts

Amplitude

The amplitude of a trigonometric function is the absolute value of the coefficient in front of the cosine or sine term. It represents the peak deviation of the function from its central value. In general, the amplitude is a measure of how 'tall' or 'short' the graph is compared to its standard form.

Period

The period of a trigonometric function is the length of the interval over which the function completes one full cycle. For functions like y = A*cos(Bx), the period is calculated as 2?/|B|, which tells us how frequently the function repeats its pattern.

Phase Shift

The phase shift indicates the horizontal displacement of the trigonometric function from its standard position. It is determined from the form y = A*cos(Bx - C) and is calculated as C/|B|, representing how far and in which direction the graph is shifted along the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions by hand involves identifying key characteristics such as the amplitude, period, and phase shift. Once these are determined, essential points within one cycle are plotted and connected smoothly to form the graph, after which the pattern is extended to complete the sketch of the function.

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