Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function. Use a graphing utility to verify your sketch. (Include two full periods.) $$y=\frac{1}{4} \cos x$$

Step-by-Step Solution

Answer

The graph of the function \(y=\frac{1}{4} \cos x\) is a cosine wave with amplitude of \(\frac{1}{4}\) and period \(2\pi\). At x-values: 0, \(2\pi\), \(4\pi\), the function hits its peak of \(\frac{1}{4}\). At \( \pi\) and \(3\pi\), it hits its lowest value of -\(\frac{1}{4}\). At \(\pi/2\), \(3\pi/2\), \(\frac{5\pi}{2}\), \( \frac{7\pi}{2}\), it hits x-axis.

Step 1: Identify the amplitude, period

Since the function is \(y=\frac{1}{4} \cos x\), the amplitude (which is the vertical stretch/ height of the graph) is \( \frac{1}{4} \). The interval of x that results in one complete wave is the period, and for the standard cosine function, the period is \(2\pi\) (or \(360^\circ\) in degrees). There are no changes in this function that will modify the period, therefore, the period is \(2\pi\).

Step 2: Plot key points

Major peaks, troughs and zero-crossings of cosine function occur at: \(x=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\). Hence, the y-values at these x-values are: \(y = \frac{1}{4}\cos(0) = \frac{1}{4}, y = \frac{1}{4}\cos(\frac{\pi}{2}) = 0, y = \frac{1}{4}\cos(\pi) = -\frac{1}{4}, y = \frac{1}{4}\cos(\frac{3\pi}{2}) = 0, y = \frac{1}{4}\cos(2\pi) = \frac{1}{4}\). Given we need two full periods of this cosine function, repeat these points for the second period by adding \(2\pi\) to the x-values.

Step 3: Sketch the graph

To sketch the graph, plot the above key points on a plane. Start with the point (0, \(\frac{1}{4}\)) then move to ( \(\frac{\pi}{2}\), 0), ( \(\pi\), -\(\frac{1}{4}\)), ( \(\frac{3\pi}{2}\), 0), and ( \(2\pi\), \(\frac{1}{4}\)), this completes one period. Now start the second period with the same pattern. After plotting the points, draw a smooth curve through the points for two periods.

Step 4: Verify with a graphing utility

Finally, to confirm your sketch, compare your graph with the graph plotted by a graphing utility, making sure they match approximate pattern, amplitude and period for two full periods of the function

Key Concepts

Amplitude of Trigonometric Functions

When you're dealing with trigonometric functions like cosine, the amplitude represents the maximum distance that the function value (the 'y' value) has from the horizontal axis (generally the x-axis). This can be visualized on a graph as the height of the peaks or the depth of the troughs from the centerline of the wave.

For the cosine function, which normally oscillates between 1 and -1, any coefficient in front of the cosine influences the amplitude. In the case of the function \(y=\frac{1}{4} \cos x\), the amplitude is \(\frac{1}{4}\). This means the graph of this function will never go higher than \(\frac{1}{4}\) or lower than \(-\frac{1}{4}\) above the horizontal axis. Hence, understanding amplitude is pivotal in predicting the extent of the wave's stretch or compression in the vertical direction.

Period of Trigonometric Functions

The period of a trigonometric function is the length of the interval to complete one full cycle of the wave on a graph. For the cosine function, the standard period is \(2\pi\) radians, as it repeats its pattern every \(2\pi\) radians.

In the function \(y=\frac{1}{4} \cos x\), there are no horizontal stretches or compressions, and thus the period remains unchanged at \(2\pi\). Recognizing the period is crucial because it guides you in knowing how far along the x-axis your graph should extend in order to sketch a complete wave. To show more than one cycle of the wave, simply continue the pattern of the wave beyond \(2\pi\), ensuring to maintain the same amplitude and shape throughout.

Sketching Trigonometric Graphs

Knowing how to sketch trigonometric graphs is essential for visualizing these functions comprehensively. As you begin sketching the graph of a cosine function, it's helpful to mark key points that represent the peak, the trough, and the points where the function crosses the horizontal axis (also called zero-crossings).

For the function \(y=\frac{1}{4} \cos x\), start by plotting the points \((0, \frac{1}{4})\), \((\frac{\pi}{2}, 0)\), \((\pi, -\frac{1}{4})\), \((\frac{3\pi}{2}, 0)\), and \((2\pi, \frac{1}{4})\). Connect these points with a smooth, periodic wave. Make sure the turns at the peaks and troughs are rounded since cosine waves have a smooth and continuous nature. Replicate these points for subsequent cycles to sketch multiple periods. With practice, sketching these graphs becomes intuitive.

Graphing Utilities for Trigonometry

Graphing utilities for trigonometry are incredibly helpful tools that can verify the accuracy of your hand-drawn sketches. These utilities allow you to enter the equation of the trigonometric function and then display the graph on screen, providing a precise visual representation of the function's behavior.

When using a graphing utility to check the function \(y=\frac{1}{4} \cos x\), you can observe that the accurately plotted graph matches your sketch in terms of amplitude, period, and the key points plotted. This is an excellent way to confirm that your understanding of amplitude and period, as well as your graph sketching skills, are on point. It's always advisable to use graphing utilities as a backup to ensure your graph reflects the true characteristics of the trigonometric function in question.