Problem 19 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of vertical stretches and compressions to graph at least two cycles of the given functions. $$g(x)=\frac{3}{2} \cos x$$

Step-by-Step Solution

Answer

The amplitude of the function is 1.5 and the period is \(2 \pi\). Two cycles of the function can be graphed by plotting the values for one cycle (0 to \(2 \pi\)) and repeating the pattern for the second cycle (\(2 \pi\) to \(4 \pi\)).

Step 1: Determine the Amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient of the cosine term. Therefore, the amplitude of \(g(x)=\frac{3}{2} \cos x\) is \( |\frac{3}{2}| = 1.5\)

Step 2: Identify the Period

The period of a cosine function with no horizontal shift or stretch is \(2 \pi\). Therefore, the period of \(\frac{3}{2}\cos x\) will remain the same as any standard cosine function, that is, \(2 \pi\).

Step 3: Graph the Function

Starting from \(x=0\), because \(\cos(0)=1\), the function will start from \(1.5 \times 1 = 1.5\). After a quarter of a period, which is \(\frac{\pi}{2}\), the function will reach its minimum because \(\cos(\frac{\pi}{2}) = 0\). Then after another quarter period, the function returns to its maximum value. This makes one complete cycle from 0 to \(2 \pi\). The same pattern is repeated for the next cycle from \(2 \pi\) to \(4 \pi\).

Key Concepts

Amplitude

The amplitude is an essential part of understanding trigonometric functions like the cosine function. In any trigonometric function, the amplitude refers to the height of the wave from the midline to its peak. For the cosine function, you calculate the amplitude by taking the absolute value of the coefficient in front of the cosine term.
For example, in the function \(g(x) = \frac{3}{2} \cos x\), the coefficient is \(\frac{3}{2}\). Therefore, the amplitude is \(|\frac{3}{2}| = 1.5\). Lastly, remember:

• The amplitude affects how tall or short the graph appears.
• It gives you insight into the maximum displacement from the middle line the wave reaches.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and it is often written as \(\cos x\). It plays a crucial role in modeling periodic phenomena such as waves. Here's what to know:

• The cosine function begins at its maximum value when \(x = 0\).
• As you move along the x-axis towards \(\pi\), it decreases to its minimum.
• Then, it returns to its starting maximum point by the time you reach \(2\pi\).

This pattern is known as a wave cycle. In our exercise, \(g(x) = \frac{3}{2} \cos x\), we observe two cycles from \(0\) to \(4\pi\). The peaks and troughs follow the same pattern, only stretched vertically due to the amplitude.

Period of a Function

The period of a function is determined by how often it completes one full cycle. For the standard cosine function, the period is \(2\pi\), as this is the interval over which it completes one full wave.
In the expression \(g(x) = \frac{3}{2} \cos x\), the period remains \(2\pi\) because there is no horizontal stretch or compression affecting it.

• The period informs you on how wide each repetitive segment of the graph will be.
• A standard cosine wave repeats its cycle every \(2\pi\) interval.

Thus, understanding the period allows you to predict the behavior of the cosine function over any interval of the x-axis.