Problem 55 - Algebra 2

Question

Find the asymptotes of the graph of each equation. $$ y=\frac{5}{2-x} $$

Step-by-Step Solution

Answer

The vertical asymptote is \( x = 2 \) and the horizontal asymptote is \( y = 0 \).

Step 1: Find the Vertical Asymptote

The vertical asymptote occurs at the \( x \) value that makes the denominator equal to zero. For the equation \( y = \frac{5}{2-x} \), set the denominator equal to zero and solve for \( x \): \( 2 - x = 0 \). Adding \( x \) to both sides gives: \( x = 2 \). So, \( x = 2 \) is the vertical asymptote.

Step 2: Find the Horizontal Asymptote

The horizontal asymptote is found by looking at the degrees of the numerator and denominator. If the degree of the denominator is greater than that of the numerator, as in this case, the horizontal asymptote is \( y = 0 \).

Key Concepts

Vertical Asymptote

A vertical asymptote on the graph of a function occurs where the function is undefined. This typically happens where the denominator of a rational function equals zero. For the given rational function, the vertical asymptote is found by setting the denominator equal to zero and solving for the variable.

• Given the function: \( y = \frac{5}{2-x} \)
• To find where the function is undefined, set the denominator \( 2-x \) equal to zero.
• When you solve \( 2-x = 0 \), you add \( x \) to both sides, which results in \( x = 2 \).

Thus, the vertical asymptote is the line \( x = 2 \). This means that as the graph approaches \( x = 2 \), the values of \( y \) will increase or decrease without bound, creating a barrier where the graph cannot actually touch or cross. Understanding vertical asymptotes is crucial as they indicate restrictions in the domain of rational functions.

Horizontal Asymptote

Horizontal asymptotes illustrate how a function behaves as the input values become very large either positively or negatively. In rational functions, these asymptotes are determined by considering the degrees of the polynomial in the numerator and the polynomial in 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 are equal, the horizontal asymptote is determined by dividing the leading coefficients.
• If the numerator's degree is greater, there is no horizontal asymptote. However, an oblique asymptote may occur.

For the equation \( y = \frac{5}{2-x} \), the numerator has a degree of 0 (constant) and the denominator has a degree of 1 (\( x^1 \)). Therefore, the horizontal asymptote is given by \( y = 0 \). This asymptote means that as \( x \) grows large in the positive or negative direction, the value of \( y \) gets closer and closer to zero.

Rational Functions

Rational functions consist of fractions where both the numerator and the denominator are polynomials. They are a key topic in algebra because they can represent real-world quantities that exhibit variable relationships.

Here are some crucial aspects of rational functions:

• The form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
• Asymptotes, both vertical and horizontal, provide insight into the behavior and limitations of the function.
• Vertical asymptotes arise when the values of \( x \) make \( Q(x) = 0 \) but not \( P(x) \).
• Horizontal asymptotes depend on the degrees of \( P(x) \) and \( Q(x) \).

Understanding rational functions is essential because they combine polynomial functions while introducing new behaviors like asymptotic approaches. This enables modeling of complex relationships seen frequently in fields such as economics, physics, and engineering.