In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)

Verified step by step guidance

1. Identify the given functions: \(f(x) = \cos x\), \(g(x) = \sin 2x\), and \(h(x) = (f - g)(x) = f(x) - g(x)\).
2. Understand that to find \(h(x)\), you subtract the value of \(g(x)\) from \(f(x)\) for each \(x\) in the interval \(0 \leq x \leq 2\pi\).
3. Create a table of values for \(x\) at key points (such as \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\)), and calculate \(f(x)\), \(g(x)\), and then \(h(x) = f(x) - g(x)\) for each point.
4. Plot the graphs of \(f(x) = \cos x\) and \(g(x) = \sin 2x\) on the same coordinate system over the interval \(0 \leq x \leq 2\pi\).
5. Using the values of \(h(x)\) from your table, plot the graph of \(h(x)\) by subtracting the \(y\)-coordinates of \(g(x)\) from those of \(f(x)\) at each \(x\), resulting in the graph of \(h\).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, such as 0 to 2π. Understanding the shape, period, amplitude, and key points of functions like cosine and sine is essential to accurately represent them on a coordinate system.

Function Operations (Addition and Subtraction)

Function operations involve combining two functions by adding or subtracting their outputs for each input value. For h(x) = (f − g)(x), this means subtracting the y-values of g(x) from f(x) at every x, resulting in a new function whose graph is derived from the pointwise difference.

Periodicity and Frequency of Trigonometric Functions

Periodicity refers to the repeating nature of trig functions. Cos x has a period of 2π, while sin 2x has a period of π due to the frequency multiplier 2. Recognizing these periods helps in understanding how the combined function h(x) behaves and repeats over the interval.