Problem 83 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=\cos \left(\frac{x}{2}\right)+\cos (2 x), 0 \leq x \leq 4 \pi$$

Step-by-Step Solution

Answer

Graph \(y = \cos \left(\frac{x}{2}\right) + \cos (2x)\) over \(0 \leq x \leq 4\pi\) by summing the y-values of each function.

Step 1: Understand the Function

We are given a function to graph: \(y = \cos \left(\frac{x}{2}\right) + \cos (2x)\). This is a sum of two cosine functions. The challenge is to add these two functions and graph the result over the interval \(0 \leq x \leq 4\pi\).

Step 2: Analyze Each Component Function

Let's break this down into the two separate functions:1. \(f(x) = \cos \left(\frac{x}{2}\right)\): This function has a period of \(4\pi\) because the standard period of \(\cos(x)\) is \(2\pi\) and here it is stretched by a factor of 2. 2. \(g(x) = \cos(2x)\): This function has a period of \(\pi\) because the standard period of \(\cos(x)\) is \(2\pi\) and it is compressed by a factor of 2.

Step 3: Sum the Functions

To graph the summed function \(y = \cos \left(\frac{x}{2}\right) + \cos (2x)\), you add the ordinates (the y-values) of the two functions at each point within the interval. This requires evaluating \(\cos \left(\frac{x}{2}\right)\) and \(\cos(2x)\) at several critical points, adding the results, and plotting these summed values.

Step 4: Graph Each Function Separately

First, sketch the graph of \(y = \cos \left(\frac{x}{2}\right)\) over \(0 \leq x \leq 4\pi\). This will be a wave with peaks and troughs every \(4\pi\). Next, sketch the graph of \(y = \cos(2x)\) over the same interval, which will have a smaller period with peaks and troughs every \(\pi\).

Step 5: Plot the Summed Graph

Now, for each important point (like multiples of \(\pi\)), add the y-values of \(\cos \left(\frac{x}{2}\right)\) and \(\cos(2x)\) and plot these points. Connect the points smoothly to illustrate the wave-like form of the summed function.

Key Concepts

Cosine Function

The cosine function, often represented as \( y = \cos(x) \), is a fundamental trigonometric function that describes a wave. It is periodic, symmetric, and oscillates between -1 and 1. The shape of the cosine graph resembles a smooth wave that starts at its maximum value, decreases to its minimum, and then returns to the maximum.

• The cosine function has a period of \( 2\pi \), which means it repeats its shape every \( 2\pi \) units along the x-axis.
• Its amplitude, or maximum height, is 1.
• It's even, meaning \( \cos(-x) = \cos(x) \).

To graph any variant of the cosine function like \( \cos\left(\frac{x}{2}\right) \) or \( \cos(2x) \), you adjust its period and amplitude. For example, \( \cos\left(\frac{x}{2}\right) \) stretches the period, making it twice as long at \( 4\pi \). In contrast, \( \cos(2x) \) compresses the period, halving it to \( \pi \).

Understanding these adjustments helps in creating accurate graphs and interpreting their behavior over specific intervals.

Period of a Function

The period of a function is the distance along the x-axis before the function starts repeating its pattern. For trigonometric functions like cosine, understanding the period is crucial for graphing.

• The regular cosine function \( y = \cos(x) \) has a period of \( 2\pi \).
• If the input, x, is adjusted, as in \( y = \cos(bx) \), the period changes to \( \frac{2\pi}{|b|} \).

For example, in the function \( \cos\left(\frac{x}{2}\right) \), the period changes to \( 4\pi \), stretching the cycle over a longer interval. Whereas for \( \cos(2x) \), the period becomes \( \pi \), compressing it to half the usual length. Recognizing these changes allows you to accurately plot the function within its correct interval, leading to clearer and precise graphs.

Summation of Functions

The summation of functions involves adding two or more functions together to form a new function. This process requires combining the y-values of each function at corresponding x-values.

• For instance, if \( f(x) = \cos\left(\frac{x}{2}\right) \) and \( g(x) = \cos(2x) \), the summation \( y = f(x) + g(x) \) means adding their y-values at each point on the x-axis.
• This summation creates a new function that might have different wave patterns or amplitudes compared to the original ones.

By graphing each original function separately, you visualize their behavior over the interval \( 0 \leq x \leq 4\pi \). Then, at critical points like \( \pi, 2\pi, \) and so on, you add \( \cos\left(\frac{x}{2}\right) \) and \( \cos(2x) \) y-values to plot the new points of the summed function. Connecting these points smoothly can highlight complex interactions and waveforms resulting from the summation. This method is essential in fields such as engineering and physics, where superimposing wave patterns is common.