Problem 53 - Algebra 2

Question

Find the asymptotes of the graph of each equation. $$ y=\frac{-1}{x+1}+1 $$

Step-by-Step Solution

Answer

The vertical asymptote of the function \(y = -\frac{1}{x+1} + 1\) is \(x = -1\) and the horizontal asymptote is \(y = 1\).

Step 1: Finding the Vertical Asymptote

To find the vertical asymptote, set the denominator equal to zero and solve for x. The denominator here is \(x+1\). So, solve \(x+1=0\) for x to get \(x = -1\). Therefore, the vertical asymptote of the function is \(x = -1\).

Step 2: Finding the Horizontal Asymptote

For the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. Here, the degree of the numerator is 0 (as it is a constant) and the degree of the denominator is 1. When the degree of the denominator is greater than the numerator, the horizontal asymptote is \(y=0\). However, we have a constant term of +1 outside of the rational fraction. This slightly changes the location of the horizontal asymptote from \(y=0\) to \(y=1\).

Key Concepts

Rational Functions

Rational functions are a special type of function represented as the quotient of two polynomials, expressed generally as \( f(x) = \frac{p(x)}{q(x)} \). These functions can exhibit interesting behaviors and features, one of which includes asymptotes. Asymptotes are lines that the graph of the function approaches but never quite touches. One of the primary characteristics of rational functions is their ability to graphically display both vertical and horizontal asymptotes.

• Numerator and Denominator: In a rational function, understanding the roles of the numerator \( p(x) \) and the denominator \( q(x) \) is key. The behavior of the function largely depends on the relationship between these two components.
• Behavior and Characteristics: As the values of \( x \) approach certain critical points, the function may approach infinity, zero, or a constant value, which lead to the discovery of asymptotes.

The example given, \( y = \frac{-1}{x+1} + 1 \), is a great showcase of rational functions with its unique expressions of both vertical and horizontal asymptotes. These asymptotes are direct results of the properties and degrees of the polynomials involved.

Horizontal Asymptote

Horizontal asymptotes represent values that a rational function approaches as \( x \) moves towards positive or negative infinity. They give insight into the end behavior of the function. The process to determine the horizontal asymptote largely depends on comparing the degrees of the polynomials in 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 degrees of both are equal, the asymptote is \( y = \frac{a}{b} \), where \( a \) is the leading coefficient of the numerator, and \( b \) is the leading coefficient of the denominator.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In \( y = \frac{-1}{x+1} + 1 \), the degree of the numerator (a constant) is 0 while the denominator is 1, leading initially to \( y = 0 \). The additional +1 adjusts the horizontal asymptote to \( y = 1 \). This change highlights a shift along the y-axis that must always be accounted for when constant adjustments appear in rational functions.

Vertical Asymptote

Vertical asymptotes occur where a rational function's denominator equals zero, resulting in undefined behavior for the function. These asymptotes often appear as vertical lines on the graph that the function approaches but cannot cross or touch.

• To find the vertical asymptote, set the denominator \( q(x) \) to zero and solve for \( x \).
• This calculation will tell you where the asymptote is located along the x-axis.

In our specific example, with \( y = \frac{-1}{x+1} + 1 \), the denominator \( x + 1 \) sets up a simple equation: \( x + 1 = 0 \). Solving it results in \( x = -1 \), revealing the vertical asymptote is at \( x = -1 \). This line reflects the boundary the function nears but never reaches as \( x \) gets closer to -1. It's crucial to find and understand these asymptotes in rational functions to analyze the limits and behaviors of their graphs effectively.