91Ó°ÊÓ

In trigonometry, the amplitude of a function is a measure of its vertical stretch or compression. For a cosine function, represented as
\(y = A \cos(B(t - h)) + k\)
, the amplitude is the absolute value of the coefficient
\(A\)
. This determines the height of the wave from its mid-line to its peak or trough.
In the given example
\(-3 + 5 \cos \left ( \frac{\pi t}{12} \right )\)
, the coefficient preceding the cosine function is
\(5\)
, thus the amplitude is
\(|5| = 5\)
. The graph will have peaks at
\(5\)
units above the mid-line and troughs at
\(5\)
units below it. Regardless of whether the coefficient is positive or negative, the amplitude is always a positive value because it's a distance.

The period of a cosine function refers to the length it takes for the function to complete one full cycle of its pattern. Normally, the cosine function
\(\cos(x)\)
completes a cycle every
\(2\pi\)
radians. When there is a multiplier
\(B\)
in the function in the form
\(y = A \cos(Bx)\)
, the period is calculated as
\(\frac{2\pi}{B}\)
.
For our example,
\(y=-3+5 \cos \left ( \frac{\pi t}{12} \right )\)
, we see the inside of the cosine function has a multiplier of
\(\frac{\pi}{12}\)
, and thus the period
\(T\)
is
\(\frac{2\pi}{(\pi/12)} = \frac{24}{\pi}\)
units. The graph of this function will repeat its pattern every
\(\frac{24}{\pi}\)
units along the t-axis.

A vertical shift in trigonometry is a transformation that moves a graph up or down without altering its shape. In the standard cosine equation
\(y = A \cos(B(t - h)) + k\)
,
\(k\)
represents the vertical shift. Positive values of
\(k\)
lift the graph upwards, while negative values lower it. Looking at our exercise
\(y=-3+5 \cos \left ( \frac{\pi t}{12} \right )\)
, the vertical shift is the constant
\(-3\)
, which indicates that the whole cosine curve is shifted
\(3\)
units downward. This transformation changes the mid-line of the cosine function from 0 to
\(-3\)
, and consequently, the maximum and minimum values are reduced by
\(3\)
units, adding a new layer of complexity to the graph's appearance.