Question

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\frac{1}{3} \cos 4 x\)

Short answer

The amplitude of the function \(g(x)=\frac{1}{3}\cos4x\) is \(\frac{1}{3}\) and the period is \(\frac{\pi}{2}\). The graph shows a horizontal compression by a factor of 4 and a vertical compression by a factor of \(\frac{1}{3}\) when compared to the parent function \(cos(x)\).

Step-by-step solution

Identify the Amplitude

The amplitude of a cosine function is determined by the absolute value of the coefficient of the cosine function. In this case, the coefficient is \(\frac{1}{3}\), thus the amplitude is \(\left|\frac{1}{3}\right| = \frac{1}{3}\).

Identify the Period

The period of the cosine function is given by the formula \(P=\frac{2\pi}{|b|}\), where 'b' is the coefficient of the variable 'x'. In the function \(g(x)=\frac{1}{3}\cos4x\), the coefficient 'b' is 4. Substituting b=4 into the formula, the period P is \(P=\frac{2\pi}{|4|}=\frac{\pi}{2}\).

Graph the Function

To graph the function \(g(x)=\frac{1}{3}\cos4x\), start with the basic cosine graph. Reflect the graph in the x-axis if the amplitude is negative (in this case it is not). Then, stretch or compress it vertically by the amplitude factor (in this case \(\frac{1}{3}\). The cosine wave, which usually has a period of \(2\pi\), should be adjusted according to the calculated period \(\frac{\pi}{2}\). To adjust, compress or stretch the graph horizontally by the period factor.

Describe the Transformation

By comparing the graph of \(g(x)=\frac{1}{3}\cos4x\) to the parent function \(cos(x)\), observe that the graph is first horizontally compressed by a factor of 4 because of the '4x', resulting in a shorter period. Then, it is vertically compressed by a factor of \(\frac{1}{3}\) because of the coefficient \(\frac{1}{3}\) before the cosine function, resulting in a smaller amplitude.

Key concepts

Amplitude of Cosine Function

The amplitude of a cosine function represents the peak deviation, or the maximum amount of displacement from the central position. In simpler words, it's the distance from the middle of the wave to the top or bottom. For the function g(x) = \frac{1}{3} \cos 4x, the coefficient in front of the cosine, \( \frac{1}{3} \), directly tells us the amplitude. Since amplitude is always positive, regardless of the sign of this coefficient, we find that the amplitude is \( \left|\frac{1}{3}\right| = \frac{1}{3} \).

When graphing, the amplitude adjusts the height of the 'waves' in the trigonometric function. A smaller amplitude, like \( \frac{1}{3} \), means that the wave will not be as tall as the standard cosine function, which has an amplitude of 1. This is a vertical compression, causing the wave to be less pronounced as compared to the parent function, which is cos(x) with a full amplitude of 1.

Period of Cosine Function

The period of a cosine function indicates how long it takes for the function to repeat its pattern. For the standard cos(x), one cycle is completed over an interval of \( 2\pi \) units. However, when the inside of the cosine function is multiplied by a factor, like 4 in \( \cos 4x \), it changes the rate at which the function repeats, hence altering its period.

To find the new period, we use the formula \( P = \frac{2\pi}{|b|} \), where 'b' is the number multiplying 'x' inside the cosine function. For g(x)=\frac{1}{3}\cos4x, we substitute 'b' with 4. Therefore, the period of this function is \( P = \frac{2\pi}{|4|} = \frac{\pi}{2} \). This shortened period means the function will repeat its cycle four times as frequently as the parent function. Graphically, this results in a horizontal compression, which squeezes the waves of the function closer together.

Transformations of Trigonometric Functions

Transformations of trigonometric functions like the cosine function involve shifting, stretching or compressing, and reflecting the graph. When given a cosine function, such as \( g(x) = \frac{1}{3} \cos 4x \), we analyze two main transformations: amplitude and period modifications, as already discussed.

In addition to these, other potential transformations include vertical shifts (up or down), and phase shifts (left or right). However, for our function \( g(x) \), there are no vertical or phase shifts because it matches the form \( A \cos(Bx) \) where A and B affect only the amplitude and period respectively. By comparing \( g(x) \) to its parent function \( \cos(x) \), we find that its graph has undergone a horizontal compression by a factor of 4 and a vertical compression by a factor of \( \frac{1}{3} \).

The net effect on \( g(x) \) is a faster oscillation due to the period change, and less pronounced 'waves' due to the decrease in amplitude. These are key aspects to remember when visualizing or sketching the graph of a transformed cosine function.