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

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1. Identify the two functions involved: \(y_1 = \cos x\) and \(y_2 = \sin 2x\). We will graph each separately over the interval \(0 \leq x \leq 2\pi\).
2. Create a table of values for \(y_1 = \cos x\) by choosing key points in the interval \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\) and calculate \(\cos x\) at these points.
3. Similarly, create a table of values for \(y_2 = \sin 2x\) at the same key points by calculating \(\sin 2x\) for each \(x\) value.
4. Add the corresponding \(y\)-coordinates from the two tables to find the values of \(y = \cos x + \sin 2x\) at each key point. This means for each \(x\), compute \(y = y_1 + y_2\).
5. Plot the points \((x, y)\) obtained from the sums on the coordinate plane and connect them smoothly to graph the function \(y = \cos x + \sin 2x\) over the interval \(0 \leq x \leq 2\pi\).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, typically one or more periods. Understanding the shape, amplitude, period, and phase shift of sine and cosine functions helps in accurately sketching their graphs.

Sum of Functions and Pointwise Addition

When adding two functions, the resulting function's value at each x is the sum of the individual function values at that x. For trigonometric functions, this means adding the y-coordinates of each function point-by-point to obtain the combined graph.

Properties of Sine and Cosine Functions

Sine and cosine functions have specific properties such as amplitude, period, and frequency. For example, sin(2x) has twice the frequency of sin(x), resulting in a shorter period. Recognizing these properties is essential for understanding how the combined function behaves.