In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + cos 2x

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1. Identify the two functions involved: \(y_1 = \cos x\) and \(y_2 = \cos 2x\). We will graph each separately first for \(0 \leq x \leq 2\pi\).
2. Create a table of values for \(y_1 = \cos x\) by choosing several \(x\) values between \$0\( and \(2\pi\) (for example, \)0\(, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), \(2\pi\)) and calculate the corresponding \)y_1$ values.
3. Similarly, create a table of values for \(y_2 = \cos 2x\) using the same \(x\) values, noting that the frequency is doubled, so the function completes two full cycles between \$0$ and \(2\pi\).
4. For each \(x\) value, add the corresponding \(y\)-coordinates from \(y_1\) and \(y_2\) to find the combined function value: \(y = \cos x + \cos 2x\).
5. Plot the points \((x, y)\) on the coordinate plane and connect them smoothly to graph the function \(y = \cos x + \cos 2x\) over the interval \(0 \leq x \leq 2\pi\).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting points based on their values at various angles, typically measured in radians. Understanding the shape, period, amplitude, and phase shift of functions like cosine helps visualize their behavior over an interval such as 0 to 2π.

Sum of Trigonometric Functions

When adding two trigonometric functions, the resulting graph is found by adding their y-coordinates point-by-point. This method combines the individual waveforms, producing a new function that reflects the superposition of the original functions' oscillations.

Period and Frequency of Cosine Functions

The period of a cosine function y = cos(kx) is given by 2π/k, indicating how often the function repeats. For y = cos x and y = cos 2x, their periods differ, affecting how their graphs combine when added, which is essential for accurately plotting the sum over 0 ≤ x ≤ 2π.