Problem 50 - Algebra 2

Question

Use a graphing calculator to graph each function in the interval from 0 to 2\(\pi .\) Then sketch each graph. $$ y=\cos x-x $$

Step-by-Step Solution

Answer

The graph of \(y = \cos x - x\) starts at (0,1), falls below the x-axis, then rises to (pi,0), again falls down and finally rises to cross the x-axis marginally before \(2\pi\).

Step 1: Graphing \(\cos x\)

First graph the cosine function \(\cos x\) from 0 to \(2\pi\). The cosine graph starts at (0,1), falls to (pi/2, 0), rises to (pi, -1), again raises to (3pi/2, 0) and finally at \(2\pi\), it is 1. This is one complete cycle of a cosine function.

Step 2: Graphing \(y = x\)

The next function to be graphed is \(y = x\). This is a linear function with a slope of 1 and passes through the origin. From 0 to \(2\pi\) (which is approximately 6.28), the function increases linearly.

Step 3: Graphing \(y = \cos x - x\)

To graph \(y = \cos x - x\), subtract the y-coordinate of the x graph from the y-coordinate of the cosine graph. To make it easier, pick some key points of x and \(\cos x\) to sketch it. It starts from (0,1), falls below the x-axis and again rises to (pi,0), falls down and then rises to cross the x-axis marginally before \(2\pi\).

Key Concepts

Graphing trigonometric functions

Graphing trigonometric functions can be a fun and insightful way to understand periodicity and wave-like behavior in mathematics. The trigonometric functions, including sine, cosine, and tangent, have distinct visual patterns that repeat over specific intervals. When using a graphing calculator, you can easily plot these functions to observe how they behave. For instance, the cosine and sine functions are continuous and smooth, and they complete one cycle within an interval of

• \(0 ext{ to }2 ext{\(\pi\)}\)
• They have maximum values of 1 and minimum values of -1.

By understanding this behavior, it becomes simpler to graph complex functions involving trigonometry since adding transformations or shifts can be handled more confidently once you're familiar with the basic shape.

Cosine function

The cosine function, written as \(y = \cos(x)\), is one of the fundamental trigonometric functions. It is useful in many aspects of both pure and applied mathematics. The cosine function represents the x-coordinate of a point on a unit circle as it travels around the circle. Here are some key characteristics of the cosine function:

• Starts at point (0,1) in its cycle.
• Completes one cycle at \(2\pi\) or approximately 6.28.
• Has maximum and minimum values of 1 and -1, respectively.
• Is symmetric about the y-axis.

When we graph the cosine function from 0 to \(2\pi\), we see the typical wave-like curve that starts from 1, decreases to -1, and then goes back to 1. This periodic nature makes cosine ideal for modeling cyclic phenomena.

Graph transformations

Graph transformations are changes applied to functions that alter their position, shape, or size on a graph. For the function \(y = \cos(x) - x\), we're transforming the basic cosine graph by subtracting the value of \(x\). This is known as a vertical translation. Here's how it works:

• Starting at each x-value, the graph of \(x\) is moved vertically.
• The shifted graph changes by the amount of \(x\) at every point.
• Subtracting \(x\) generally results in lowering the entire graph depending on value of \(x\).

Visualizing these transformations helps in predicting where new intersections and shifts might occur, which is crucial for sketching accurate graphs.

Graphing intervals

Graphing intervals define the range over which a particular graph should be plotted. By focusing on specific intervals, we can glean important insights from complex equations. In this case, we're interested in graphing functions between \(0 \text{ and } 2\pi\).Using this interval is especially helpful when dealing with trigonometric functions like cosine since one full period is captured in \(0 \text{ to } 2\pi\). This assures that the entire behavior of the function, including all characteristic peaks, troughs, and zero crossings, can be observed. For completions:

• It helps in comparing trigonometric graphs over the same specified range without distractions of longer intervals.
• Allows easy identification of periodicity and pivotal intersections with axes as seen in case-specific tasks like graphing \(y = \cos x - x\).

Accurate interval selection is essential for effective graphical analysis and function comparison.