Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Provide a short answer to each question. Do not use a calculator. What is the equation of the vertical asymptote of the graph of \(y=\frac{1}{x-3}+2 ?\) of the horizontal asymptote?

Step-by-Step Solution

Answer

Vertical asymptote: \(x=3\). Horizontal asymptote: \(y=2\).

Step 1: Identify the Vertical Asymptote

The vertical asymptote occurs where the function is undefined. For the given function:\[y = \frac{1}{x-3} + 2\]the term \(\frac{1}{x-3}\) becomes undefined when the denominator equals zero, i.e.,\[x-3=0\]Thus, solving this equation for \(x\) gives us the vertical asymptote at \(x=3\).

Step 2: Determine the Horizontal Asymptote

The horizontal asymptote of a function in the form \(y = \frac{a}{b(x-c)} + d\) is usually affected by the constant term outside the fraction. In this function:\[y = \frac{1}{x-3} + 2\]the horizontal asymptote is \(y = 2\), as this is the line the function approaches as \(x\) tends towards infinity.

Key Concepts

Vertical Asymptote

In precalculus, a vertical asymptote defines a line that a graph approaches but never touches or crosses. It's like an invisible barrier at which the function becomes undefined. For rational functions like \( y = \frac{1}{x-3} + 2 \), the vertical asymptote can be found by determining where the denominator of the fraction equals zero.

• First, take the denominator \( x - 3 \) of the rational part of the function.
• Set it equal to zero, resulting in the equation \( x - 3 = 0 \).
• Solve for \( x \) to find \( x = 3 \).

Thus, the vertical asymptote is at \( x = 3 \). This means that as \( x \) approaches 3 from both sides, the value of \( y \) will increase or decrease towards infinity, reflecting an endless climb or dive along the graph.

Horizontal Asymptote

Horizontal asymptotes represent a line that the graph approaches and flattens towards as the input value \( x \) goes to positive or negative infinity. This is especially important in rational functions, where the numerators and denominators play a key role.

For the function given, \( y = \frac{1}{x-3} + 2 \), the horizontal asymptote is heavily influenced by the constant term added to the fraction. As \( x \) becomes very large or very small, the value of \( \frac{1}{x-3} \) tends towards zero. What remains significant is the constant \( +2 \). Here's a step-by-step breakdown:

• As \( x \to \infty \) or \( x \to -\infty \), the fraction \( \frac{1}{x-3} \to 0 \).
• The function simplifies to approximately \( y = 0 + 2 \), which is just \( y = 2 \).
• Therefore, as \( x \) moves towards infinity, the function levels off at this horizontal line.

Thus, the horizontal asymptote of the function is the line \( y = 2 \). This is the value that \( y \) will never quite reach but will get closer and closer to indefinitely.

Graphing Rational Functions

Graphing rational functions involves understanding both vertical and horizontal asymptotes, along with intercepts and general shape. For a function like \( y = \frac{1}{x-3} + 2 \), knowing where it behaves asymptotically provides a reliable blueprint for plotting.

Here's how to begin graphing this function:

• Identify vertical and horizontal asymptotes: We've found that our vertical asymptote is at \( x = 3 \) and our horizontal asymptote at \( y = 2 \).
• Check for intercepts: Set \( x = 0 \) to find \( y \)-intercept and set \( y = 0 \) to find \( x \)-intercepts if they exist. For this function, the calculation can show that there is no intersection with the x-axis.
• Study the behavior near the asymptotes: As \( x \) approaches 3, the function will shoot off towards infinity or negative infinity. As \( x \to \infty \), \( y \) will approach the constant 2.
• Plot a few key points to define the graph's curve, such as points near the asymptotes and intercepts.

These steps together give you a comprehensive understanding of the behavior and appearance of the rational function's graph. It'll help immensely in visualizing and analyzing how these functions react as conditions change.