Problem 78 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=3 \sin x-\cos x, 0 \leq x \leq 2 \pi$$

Step-by-Step Solution

Answer

Plot the function \(y = 3 \sin x - \cos x\) over the interval using critical points: \((-1, 3, 1, -3, -1)\).

Step 1: Understanding the Functions

The given functions to add are \(y_1 = 3 \sin x\) and \(y_2 = -\cos x\). We will sum their ordinates (i.e., their values at specific points) to find the function \(y = 3 \sin x - \cos x\) over the interval \(0 \leq x \leq 2\pi\).

Step 2: Evaluate Special Angles

Consider common angle values for sine and cosine: \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\). Calculate: - At \(x=0, y = 3\sin(0) - \cos(0) = 0 - 1 = -1\).- At \(x=\frac{\pi}{2}, y = 3\sin(\frac{\pi}{2}) - \cos(\frac{\pi}{2}) = 3 \cdot 1 - 0 = 3\).- At \(x=\pi, y = 3\sin(\pi) - \cos(\pi) = 0 - (-1) = 1\).- At \(x=\frac{3\pi}{2}, y = 3\sin(\frac{3\pi}{2}) - \cos(\frac{3\pi}{2}) = -3\cdot1 - 0 = -3\).- At \(x=2\pi, y = 3\sin(2\pi) - \cos(2\pi) = 0 - 1 = -1\).

Step 3: Plot the Graph

Plot the calculated points on a graph over the interval \(0 \leq x \leq 2\pi\) using the values determined in Step 2. Connect these points considering the periodic behavior of the sine and cosine functions to form a smooth curve depicting \(y = 3\sin x - \cos x\).

Key Concepts

graphing functions

Graphing functions can be seen as a way of visually representing the behavior and patterns that a function follows over a specific interval. In the exercise above, we are tasked with plotting the function \(y = 3\sin x - \cos x\) over the interval \(0 \leq x \leq 2\pi\). To do this, we start by determining the values of \(y\) at certain key points within this interval, such as \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \(2\pi\).

These points are often chosen because they represent significant changes in the characteristic behavior of the trigonometric functions \(\sin x\) and \(\cos x\). By calculating these critical values and plotting them on a graph, we get a better picture of how the function behaves across the interval. Once plotted, connecting these points by smooth curves helps to illustrate the continuous nature of trigonometric functions.

• Use key points to plot the function.
• Assess the behavior using intervals of \(0 \leq x \leq 2\pi\).
• Connect the points smoothly to visualize the function.

sine and cosine

Sine and cosine are fundamental trigonometric functions that play a crucial role in various mathematical domains, including geometry, physics, and engineering. They are periodic functions that repeat their values in a predictable manner. In the exercise, we deal with \(3\sin x\) and \(-\cos x\), understanding that \(\sin x\) varies from \(-1\) to \(1\) as \(x\) changes from \(0\) to \(2\pi\), while \(\cos x\) operates the same in terms of output range.

In this problem, we apply a multiplier to the \(\sin x\), making it \(3\sin x\). This changes its peak values from \(1\) and \(-1\) to \(3\) and \(-3\), leading to a vertically stretched graph. Meanwhile, \(-\cos x\) flips the cosine graph upside down, affecting its overall appearance but not its amplitude. These alterations are essential in shaping the exact curve of \(y = 3\sin x - \cos x\).

• Sine and cosine have outputs between \(-1\) and \(1\).
• Multipliers affect amplitude, changing peak heights.
• Negative signs can invert the graph direction.

periodic behavior

The periodic behavior of functions like sine and cosine means they repeat their patterns at regular intervals, known as the period. For both \(\sin x\) and \(\cos x\), the period is \(2\pi\). This indicates that every \(2\pi\) units, the function completes its cycle and starts anew. Understanding this cyclic nature is crucial when summing or modifying these functions as we've done in the exercise.

The result for the given function, \(y = 3\sin x - \cos x\), is a new function that also demonstrates periodic behavior, though its specific properties like peaks, troughs, and zero crossings might vary depending on the coefficients of \(\sin x\) and the transformation applied to \(\cos x\). Recognizing periodic traits aids in predicting the function's behavior beyond the plotted interval. It also helps determine important characteristics like frequency and symmetry.

• Trigonometric functions are periodic with a period of \(2\pi\).
• Periodic functions repeat values regularly.
• Altering these functions maintains the periodicity but may change the details of its cycles.