Problem 6 - Contemporary Precalculus

Question

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=\frac{1}{x}$$

Step-by-Step Solution

Answer

Question: Sketch the graph of the equation y = 1/x using six chosen points, and describe its behavior around these points. Answer: For the equation y = 1/x, we chose six points for x: -2, -1, 1/2, 1, 2, and 3. Their corresponding y-values are -1/2, -1, 2, 1, 1/2, and 1/3 respectively. After plotting these points, we observed that as x approaches 0 from the left, y decreases toward negative infinity, while as x approaches 0 from the right, y increases toward positive infinity. Additionally, as x tends to positive or negative infinity, y approaches 0. The graph has a vertical asymptote at x=0 and a horizontal asymptote at y=0.

Step 1: Choose six points

We'll choose the following six points to graph the equation y = 1/x: 1. x = -2 2. x = -1 3. x = 1/2 4. x = 1 5. x = 2 6. x = 3

Step 2: Evaluate y for each point

Let's find the corresponding value of y for each of our chosen x points: 1. x = -2 : y = 1/-2 = -1/2 2. x = -1 : y = 1/-1 = -1 3. x = 1/2 : y = 1/(1/2) = 2 4. x = 1 : y = 1/1 = 1 5. x = 2 : y = 1/2 6. x = 3 : y = 1/3 = 1/3

Step 3: Plot the points

Plot the six points on the graph: 1. (-2, -1/2) 2. (-1, -1) 3. (1/2, 2) 4. (1, 1) 5. (2, 1/2) 6. (3, 1/3)

Step 4: Analyze the behavior of the graph

Once we've plotted our points on the graph, we can observe the behavior of the graph around these points. We notice that as x approaches 0 from the left (negative values), y tends to decrease toward negative infinity, and as x approaches 0 from the right (positive values), y tends to increase toward positive infinity. We also notice that as x tends to positive or negative infinity, y approaches 0. This means that the graph has a vertical asymptote at x=0 and a horizontal asymptote at y=0.

Step 5: Fill in the rest of the graph

Using our observation from the previous step, we can now sketch the rest of the graph by connecting the plotted points. Make sure to draw the two asymptotic lines at x=0 and y=0 as dashed lines to indicate that the graph never crosses these lines.

Step 6: Verify with a calculator

Now, utilize a calculator or graphing software to graph the equation y = 1/x and compare the results with our manually drawn graph. The calculator's graph should match our graph, confirming that we've sketched the graph accurately using the six points we chose.

Key Concepts

Vertical Asymptote

Understanding vertical asymptotes is crucial when graphing rational functions. A vertical asymptote occurs where the function's denominator equals zero, resulting in division by zero, which is undefined. For the function \(y = \frac{1}{x}\), the denominator \(x\) becomes zero when \(x = 0\).
The graph of a function will approach this line closely but never touch or cross it. Vertical asymptotes are depicted as dashed lines on graphs. The behavior of the graph near the vertical asymptote can be dramatically different depending on the direction from which \(x\) approaches the asymptote line. In this exercise, as \(x\) approaches 0 from the right, \(y\) becomes very large (tending towards infinity). Conversely, as \(x\) approaches 0 from the left, \(y\) decreases (tending towards negative infinity).
Recognizing the vertical asymptote is essential for accurately sketching the graph, ensuring that the behavior around \(x = 0\) is captured correctly.

Horizontal Asymptote

A horizontal asymptote occurs when the values of \(y\) tend to approach a constant number as \(x\) goes to positive or negative infinity. For the rational function \(y = \frac{1}{x}\), the horizontal asymptote is \(y = 0\). This indicates that as \(x\) increases or decreases indefinitely, the value of \(y\) comes closer and closer to zero.This means that even though the graph moves further along the x-axis, it will never actually touch y = 0.
In practical terms, the horizontal asymptote provides a line that the graph of a function will get arbitrarily close to, but remain separate.
Horizontal asymptotes are significant as they describe the end behavior of rational functions and are drawn as dashed lines to indicate where the graph will not cross. Understanding horizontal asymptotes allows you to predict how the graph behaves at extreme values of \(x\), which is a powerful tool in graphing.

Manual Graphing Techniques

Graphing rational functions manually can be a valuable skill, providing deeper insight into how graphs behave. Here's a straightforward method using the example \(y = \frac{1}{x}\):

• Firstly, choose specific values of \(x\) to evaluate \(y\). Use points that simplify calculations, such as \(-2, -1, \frac{1}{2}, 1, 2, 3\).
• Calculate \(y\) for each of these \(x\)-values to get ordered pairs. Plot these points on a coordinate plane.
• Identify any asymptotes. In this case, there's a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Once you've plotted these points, connect them smoothly, aiming to follow the trend exhibited by the points and approach the asymptotic lines without crossing them.
These techniques help in building a sketch that accurately represents the function's behavior, preparing you to compare your sketch with technology-aided graph results for verification.