Problem 58 - Essential Calculus

Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x-3}{x-2} $$

Step-by-Step Solution

Answer

The function has a y-intercept at -1.5 and x-intercept at 3, a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. It does not display any symmetry to neither the x-axis nor the y-axis. With these points and asymptotes we can sketch the graph of the function.

Step 1: Identify the Intercepts

The y-intercept is the value of y when x = 0. If you substitute x = 0 in the equation \(y = \frac{x-3}{x-2}\), the y-intercept y = -3/2. The x-intercept is the value of x for which y = 0. If you substitute y = 0 in the equation, you get x = 3 as the x-intercept.

Step 2: Identify the Asymptotes

To find the vertical asymptote, set the denominator equal to zero and solve for x, which gives x = 2. This implies that the line x = 2 is a vertical asymptote. As for the horizontal asymptote, since the degree of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the coefficients of the highest powers of x in the numerator and denominator, or y = 1.

Step 3: Analyze the Symmetry

Since neither replacing \(-x\) for x nor replacing \(-y\) for y results in the original equation, there is no symmetry to neither the x-axis nor the y-axis. Thus, the function is neither even nor odd.

Step 4: Graph the function

Using the y-intercept (-1.5), x-intercept (3), vertical asymptote (x = 2), and horizontal asymptote (y = 1), sketch the graph. The graph should have two distinct parts divided by the vertical asymptote, approaching the asymptotes in the limit, but never touching them. Use a graphing utility to verify your result.

Key Concepts

Intercepts

When graphing rational functions like the one given by the equation \(y = \frac{x-3}{x-2}\), identifying intercepts is crucial. Intercepts are the points where the graph crosses the axes, either the x-axis or the y-axis.

First, let's find the y-intercept. This is the value of \(y\) when \(x = 0\). By substituting \(x = 0\) into the equation, the y-intercept can be calculated as:

• \(y = \frac{0-3}{0-2} = \frac{-3}{-2} = -1.5\)

The y-intercept is at (0, -1.5).

Next, we determine the x-intercept, where the value of \(y\) is zero. Solving for \(x\) when \(y = 0\) results in:

• \(0 = \frac{x-3}{x-2}\)
• \(x - 3 = 0\)
• \(x = 3\)

The x-intercept happens at (3, 0). These intercepts provide key points to help shape the graph of the function.

Asymptotes

In graphing rational functions, asymptotes are lines that the graph approaches but never actually touches. They are important in understanding the graph's behavior over its domain.

For the function \(y = \frac{x-3}{x-2}\), we need to find both vertical and horizontal asymptotes.

• Vertical Asymptote: Occurs where the denominator is zero. Setting the denominator \(x - 2 = 0\) and solving for \(x\), we find that \(x = 2\) is a vertical asymptote. The graph will come infinitely close to both sides of the line \(x = 2\), but never cross it.


• Horizontal Asymptote: This relates to the limits as \(x\) approaches infinity. Here, the degrees of the numerator and denominator are the same (both 1), so the horizontal asymptote is the ratio of the leading coefficients. Thus, \(y = \frac{1}{1} = 1\) is the horizontal asymptote.

These asymptotes help us understand where the graph extends into infinity and how it flattens out.

Symmetry

Symmetry in graphs can simplify understanding and drawing them, showing identical parts on either side of a central line or point. A function may exhibit symmetry about the y-axis, the x-axis, or the origin. However, the function \(y = \frac{x-3}{x-2}\) does not display any such symmetry.

To check for symmetry, you can perform these tests:

• Y-axis Symmetry: Replace \(x\) with \(-x\) and see if the equation remains unchanged. Here, substituting \(-x\) results in \(y = \frac{-x-3}{-x-2}\), which is different from the original equation.
• X-axis Symmetry: Replace \(y\) with \(-y\), which doesn't hold because it changes the equation fundamentally.
• Origin Symmetry: Replace both \(x\) and \(y\) with \(-x\) and \(-y\) respectively, which also doesn't satisfy symmetry conditions for this function.

Since it fails these tests, the function is neither even nor odd, leading us to conclude there is no symmetry in its graph.

Graphing Utility

Graphing utilities are invaluable in visualizing mathematical equations and verifying manual calculations. When you use a graphing utility with the function \(y = \frac{x-3}{x-2}\), you can observe the accuracy of your intercepts and asymptotes.

Here's how to effectively use a graphing utility for rational functions:

• Input the function equation into the utility. Ensure it's accurate.
• Zoom in/out to capture all key features such as intercepts and asymptotic behavior.
• Spot check the plot with the y-intercept at (-1.5) and the x-intercept at (3).
• Verify the vertical asymptote's effect by observing how the function draws near the line \(x = 2\), but remains apart.
• Observe the horizontal asymptote \( y = 1 \) as the function levels off at this value for large positive or negative \( x \).

This utility not only corroborates your hand-drawn sketches but deepens your insight into the function's behavior and structure.