Problem 61 - Algebra 2

Question

What are the asymptotes of the graph of \(y=\frac{2 x-1}{x+15}\).

Step-by-Step Solution

Answer

The vertical asymptote is \(x = -15\) and the horizontal asymptote is \(y = 2\).

Step 1: Finding the Vertical Asymptote

Set the denominator of the equation, \(x + 15\), equal to zero and solve for \(x\). This gives \(x = -15\). This is the equation for the vertical asymptote.

Step 2: Finding the Horizontal Asymptote

Compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \(y = \frac{a}{c}\). If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. In this case, both the numerator and denominator are the same degree, so divide the leading coefficients of the numerator and denominator. This gives \(y = \frac{2}{1} = 2\). This is the equation for the horizontal asymptote.

Key Concepts

Vertical Asymptote

A vertical asymptote occurs when the denominator of a rational function is equal to zero and the numerator is not zero at the same point. This results in the graph of the function approaching a vertical line but never actually touching it. The function can become infinitely large or infinitely small as it nears the vertical asymptote. To find a vertical asymptote, you set the denominator of the rational function to zero and solve for the variable. For example, in the function \(y = \frac{2x - 1}{x + 15}\), we set \(x + 15 = 0\) which gives \(x = -15\). The line \(x = -15\) is the vertical asymptote where the graph of the function will approach but never intersect.

Horizontal Asymptote

Horizontal asymptotes provide information about the behavior of a function as the input variable approaches positive or negative infinity. It describes a line that the graph of a function approaches, but may not actually reach. To find a horizontal asymptote for a rational function, you must compare the degrees of the polynomial in the numerator and the denominator.

• If the degree of the numerator is less than that of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, the horizontal asymptote is \(y = \frac{a}{c}\), where \(a\) and \(c\) are the leading coefficients of the numerator and the denominator, respectively.
• If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.

In the given function \(y = \frac{2x - 1}{x + 15}\), both the numerator and denominator are of the same degree, 1. Thus, the horizontal asymptote is determined by dividing the leading coefficients: \(y = \frac{2}{1} = 2\).

Rational Functions

Rational functions are quotients of two polynomials. They are expressed in the form \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\). These functions can have different types of asymptotes, which are straight lines that the graph approaches but does not cross.Rational functions can possess:

• Vertical asymptotes, where there is a division by zero in the function.
• Horizontal asymptotes, determined by the relative degrees of the numerator and denominator polynomials.
• Oblique asymptotes, which occur when the degree of the numerator is exactly one greater than the degree of the denominator (though not present in the example problem).

Understanding these asymptotes aids in graphing rational functions, as they define the boundaries within which the function operates. By carefully evaluating the behavior of the rational function, students can predict how the function behaves at extremes and around specific critical points.