Problem 7 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=\frac{3}{4} \cos \frac{x}{2} $$

Step-by-Step Solution

Answer

The amplitude of the function is \( \frac{3}{4} \) and the period of the function is \( 4\pi \)

Step 1: Identify the Amplitude

In our equation, \( y=\frac{3}{4} \cos \frac{x}{2} \), the absolute value of the coefficient of cosine gives the amplitude. So, the amplitude is \( |\frac{3}{4}| = \frac{3}{4} \)

Step 2: Identify the period

The coefficient of x inside the cosine function gives us the value of B, which can be used to find the period. The period P of a cosine function in the form \( y=A \cos (Bx) \) is given by \( P=\frac{2\pi}{|B|} \). In our equation, \( B=\frac{1}{2} \), so \( P=\frac{2\pi}{|\frac{1}{2}|}= 4\pi \)

Key Concepts

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function is a measure of its 'height', or the distance from the midpoint of the wave to its peak or trough. In the context of the cosine function, the amplitude is represented by the coefficient in front of the cosine. Consider the exercise where we are asked to find the amplitude of the function
\( y=\frac{3}{4} \cos \frac{x}{2} \).

Similarly, in the function
\( y=A\cos(Bx+C)+D \) where
\(A\), \(B\), \(C\), and \(D\) are constants, the amplitude is \( |A| \). This represents the maximum vertical displacement from its midline. Therefore, a larger value of \(A\) indicates a greater amplitude.

Given the function
\( y=\frac{3}{4} \cos \frac{x}{2} \),
the amplitude is simply the absolute value of \(\frac{3}{4}\), which is
\frac{3}{4} (as the amplitude is always taken as a positive value). This tells us that the height of the wave from its centerline to a peak (or to a trough) is \(\frac{3}{4}\) units.

Period of Cosine Function

In trigonometry, the period of a function refers to the length it takes for the function to repeat its pattern. For the cosine function, which is periodic by nature, this is the horizontal length from one peak to the next (or from any point on the graph to the next identical point). The general form of a cosine function is
\( y=A\cos(Bx+C)+D \) where \(B\) affects the period of the function.

The formula to find the period \(P\) of a cosine function is
\( P=\frac{2\pi}{|B|} \),
where \(2\pi\) is the period of the standard \(\cos(x)\) function. This formula allows us to calculate the period regardless of the function's transformations. In the given function
\( y=\frac{3}{4} \cos \frac{x}{2} \),
the coefficient of \(x\) within the cosine, \(\frac{1}{2}\), represents \(B\). Using
\( P=\frac{2\pi}{|B|} \),
we get the period as
\( P=\frac{2\pi}{|\frac{1}{2}|}=4\pi \).
This means that the cosine curve completes one full cycle over an interval of \(4\pi\) units along the x-axis.

Trigonometric Function Transformations

Transformations of trigonometric functions are changes to the function's graph that adjust its shape, position, or orientation. In the context of the cosine function, these transformations can be captured within the general equation
\( y=A\cos(Bx+C)+D \).

Each constant – \(A\), \(B\), \(C\), and \(D\) – has a distinct effect:

• \(A\) modifies the amplitude, as discussed earlier.
• \(B\) affects the period of the function, scaling the graph horizontally.
• \(C\) is a horizontal shift, moving the graph left or right, which is also known as phase shift.
• \(D\) represents the vertical shift, moving the graph up or down, shifting the midline.

The calculations in the original exercise focus on \(A\) and \(B\), determining the amplitude and period respectively, but it's important to be mindful that other transformations could also be applied to the function, changing how it's represented on a graph. Understanding these transformations is key to mastering trigonometry and analyzing sinusoidal functions in various applications.