Problem 73 - Algebra and Trigonometry

Question

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of \(h\) by adding or subtracting the corresponding \(y\)-coordinates on the graphs of \(f\) and \(g\) $$f(x)=\sin x, g(x)=\cos 2 x, h(x)=(f-g)(x)$$

Step-by-Step Solution

Answer

The graphs of the three functions \(f(x)=\sin x, g(x)=\cos 2 x\), and \(h(x)=\sin x - \cos 2x\) in the same rectangular coordinate system shows that the frequency of oscillations for \(g(x)\) is twice as much as that for \(f(x)\), and the third function, \(h(x)\), consists of modified oscillations resulting from the difference of y-coordinates between \(f(x)\) and \(g(x)\) at corresponding x-values on the graphs of \(f(x)\) and \(g(x)\).

Step 1: Plotting the first function

Firstly graph the function \(f(x) = \sin x\) on the domain \(0 \leq x \leq 2 \pi\). This graph oscillates between -1 and 1, crossing \(y=0\) at multiples of \( \pi \).

Step 2: Plotting the second function

Then graph the function \(g(x) = \cos 2x\). As this function is the cosine of twice the input (i.e., the frequency has been doubled), it oscillates between -1 and 1 twice as frequently as \(f(x)\) within the same domain.

Step 3: Calculating and plotting the third function

Given that \(h(x)=(f-g)(x)= \sin x - \cos 2x\), subtract the y-coordinates of the graph of \(g(x)\) from those of \(f(x)\) for each x-value in the domain, and plot these points to obtain \(h(x)\).

Step 4: Drawing the final graph

Finally, draw the functions \(f, g,\) and \(h\) in the same rectangular coordinate system, clearly distinguishing them from one another.

Key Concepts

Amplitude and Period of Trig Functions

Understanding the amplitude and period of trigonometric functions is crucial for graphing these functions properly.

The \textbf{amplitude} of a trigonometric function is the measure of its maximum displacement from the central axis (baseline). For instance, with the function f(x) = \(sin x\), the amplitude is 1 because the function varies from -1 to 1. When graphing, this means the highest and lowest points of the graph are 1 unit above and below the baseline, respectively.

The \textbf{period} of a trigonometric function represents the length of one complete cycle of the function. The period of f(x) = \(sin x\) is \(2\pi\) because the function repeats every \(2\pi\) units along the x-axis. Yet, for g(x) = \(cos 2x\), the function's frequency is doubled, so its period is halved. Therefore, its period is \(\pi\), and it completes one full cycle from 0 to \(\pi\).

When \textbf{graphing trigonometric functions}, it's essential to mark these two properties accurately. For the given exercise, the graphs of f(x) and g(x) should reflect their respective amplitudes and periods before any combinations of functions are considered.

Combinations of Functions

Combining functions is a method that allows the creation of new functions using basic function operations. In our case, this involves adding and subtracting the y-coordinates of two functions, f(x) and g(x), at each point along the x-axis to produce a third function, h(x).

When \textbf{combining trigonometric functions} like sine and cosine, particularly when they have different periods or frequencies, the resulting graph can become complex. The key is to carefully plot the points for each original function over the common domain. In the exercise, for every x-value from 0 to \(2\pi\), you calculate h(x) = f(x) - g(x) = \(sin x - cos 2x\), and then graph these results.

The new function might not exhibit simple periodic behavior if the periods of the original functions don't have a common multiple within the given domain. Therefore, while f(x) and g(x) have clear cycles, h(x) might not, which requires thorough evaluation and graphing of points to accurately represent h(x).

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable where both sides of the equation are defined. They are the backbone of simplifying expressions and solving trigonometric equations. Common identities include Pythagorean identities, angle sum and difference identities, and double angle identities.

In our exercise context, knowing that sin²(x) + cos²(x) = 1, one of the Pythagorean identities, can help to understand the behavior of sine and cosine functions when graphing. These identities are also instrumental when combining functions, as they can often simplify the resulting expression.

However, when subtracting functions—as in h(x) = f(x) - g(x)—direct application of these identities may not simplify the result, but they provide a deeper understanding of the relationship between the functions. As identities hold true for any value within the functions' domains, they ensure that the foundational relationships between trigonometric functions remain consistent, regardless of how they are combined or manipulated in exercises like this.