Problem 48 - Calculus: One and Several Variables

Question

Sketch the graph of the function. $$f(x)=\frac{1}{3} \cos 2 x$$

Step-by-Step Solution

Answer

The graph of \(f(x) = \frac{1}{3} cos(2x)\) is a cosine wave with amplitude 1/3 and period \(\pi\). There's no phase shift or vertical shift.

Step 1: Identify the Amplitude

Amplitude is the peak value of the function. It's the absolute value of the coefficient of the cosine function. In this case the amplitude is 1/3.

Step 2: Identify the Period

The period of the cosine function is determined by coefficient of \(x\). Period is usually \(2\pi\) but in this case, it's \(\pi\), because the coefficient of \(x\) is 2.

Step 3: Identify the Phase Shift

Phase shift is a horizontal shift, which in this case is none since there's no additional constant that \(x\) is multiplied by.

Step 4: Identify any Vertical Shift

Vertical Shift is a up or down shift. In this problem, there's no vertical shift, because there's no constant being added or subtracted from the function.

Step 5: Draw the Graph

Sketch the cosine curve, making sure to indicate the amplitude of 1/3, the period of \(\pi\), and no phase shift or vertical shift.

Key Concepts

Amplitude

The amplitude of a trigonometric function is an important concept because it tells us how much the graph stretches or compresses vertically. In any standard cosine or sine function, the amplitude is the absolute value of the coefficient in front of the cosine or sine. For the function given in the exercise, which is \[f(x) = \frac{1}{3} \cos(2x)\]The coefficient in front of the cosine is \(\frac{1}{3}\). Therefore, the amplitude is \(\frac{1}{3}\). This indicates that the peaks and valleys of the cosine wave reach only up to \(\frac{1}{3}\) above and below the midline of the graph, respectively. Understanding amplitude helps us see the extent to which a function deviates from its central axis.

Period of a Function

The period of a trigonometric function is the horizontal length of one complete cycle of the wave. For sine and cosine functions, the standard period is \(2\pi\), meaning it takes a distance of \(2\pi\) along the x-axis to see the wave repeat itself. However, if the function has a coefficient in front of \(x\), it modifies the period. The formula for the period is \[\text{Period} = \frac{2\pi}{b}\]where \(b\) is the coefficient of the variable \(x\). For the function \[f(x) = \frac{1}{3} \cos(2x)\]The coefficient \(b\) is 2. So, the period becomes \[\frac{2\pi}{2} = \pi\]This means the function will complete a full cycle over an interval of \(\pi\) instead of \(2\pi\). This is critical for determining how the graph lays out horizontally.

Phase Shift

Phase shift represents the horizontal movement of a trigonometric graph along the x-axis. A phase shift happens if there is a constant added or subtracted inside the function's argument with the variable \(x\). It helps us determine if the starting point of the graph has been moved left or right from its original position. For our example function, \[f(x) = \frac{1}{3} \cos(2x)\]there is no additional constant inside the cosine function's argument. Thus, there is no phase shift in this function. Without a phase shift, the graph starts at the typical point where the cosine function begins at \(x = 0\). Incorporating phase shifts properly can enhance your understanding of how such shifts impact the graph's orientation.

Vertical Shift

Vertical shift entails moving the entire graph of the function up or down along the y-axis. This occurs when a constant is added or subtracted to the function as a whole. It can change the midpoint line around which the wave oscillates.In the equation provided \[f(x) = \frac{1}{3} \cos(2x)\]no constant is added or subtracted outside of the cosine function. Because of this, there is no vertical shift; the graph oscillates symmetrically around the x-axis. Vertical shifts, when present, allow a wave to position itself higher or lower on the graph, thus altering its relation to the axis without affecting amplitude or period.