Problem 5 - A Graphical Approach to College Algebra

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: Identifying Vertical Asymptote

A vertical asymptote occurs in rational functions when the denominator is equal to zero because the function becomes undefined. For the given function \(y = \frac{1}{x-3} + 2\), we set the denominator equal to zero: \(x - 3 = 0\). Solving for \(x\), we find \(x = 3\). Thus, the equation of the vertical asymptote is \(x = 3\).

Step 2: Identifying Horizontal Asymptote

The horizontal asymptote of a rational function depends on the degrees of the numerator and denominator. Since the highest degree of \(\frac{1}{x-3}\) is zero (constant term), the horizontal asymptote occurs when \(x\) approaches infinity. In this case, the function approaches the value of the constant added outside the fraction, which is \(2\). Therefore, the horizontal asymptote is \(y = 2\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are found in rational functions and occur at values of \(x\) which make the denominator zero—this means that the function is undefined at these points. For the function \(y = \frac{1}{x-3} + 2\), we find the vertical asymptote by setting the denominator \(x - 3\) equal to zero and solve for \(x\).
We perform the following steps to find the vertical asymptote:

• Set the denominator equal to zero: \(x - 3 = 0\).
• Solve for \(x\): this yields \(x = 3\).

Hence, the vertical asymptote is given by the equation \(x = 3\).
This means that as \(x\) approaches 3 from either direction, the function values grow very large in positive or negative direction, but they can never actually reach or cross the asymptote. Keep in mind, vertical asymptotes are not part of the graph of the function.

Horizontal Asymptotes

Horizontal asymptotes describe how a function behaves as \(x\) approaches infinity or negative infinity.
These asymptotes are significant in understanding the end behavior of a function.
For the rational function \(y = \frac{1}{x-3} + 2\), we determine the horizontal asymptote by considering the degrees of the numerator and the denominator. Since the numerator \(1\) is a constant and the denominator \(x-3\) is a linear term, the function \(\frac{1}{x-3}\) decreases towards zero as \(x\) becomes very large or very small.
Concretely, as \(x\) approaches either positive or negative infinity, \(\frac{1}{x-3}\) approaches 0. However, because of the additional constant term \(+2\), the function approaches the horizontal line \(y = 2\).
This tells us that no matter how large or small \(x\) becomes, the outputs of the function will gravitate closely to 2, thereby defining the horizontal asymptote \(y = 2\).
Horizontal asymptotes can be crossed by the graph at finite points, unlike vertical asymptotes.

Rational Functions

Rational functions are any functions expressed as the ratio of two polynomials, typically in the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
They are noted for potentially having asymptotes, which are lines that the graph of the function approaches, but does not necessarily intersect.
Leaning on the example of \(y = \frac{1}{x-3} + 2\):

• The numerator is \(1\), a constant.
• The denominator is \(x-3\), a simple linear polynomial.
• The resulting rational function can have both vertical asymptotes when the denominator equals zero and horizontal asymptotes determined by the behavior of \(\frac{1}{x-3}\) at the extremes of \(x\).

Such functions often exhibit discontinuities at vertical asymptotes, whereas horizontal asymptotes provide a simplified view of their behavior at infinity.
Understanding these concepts helps in graphing and analyzing rational functions for better comprehension.