Problem 19 - Algebra 2

Question

Find the horizontal asymptote of the graph of each rational function. $$ y=\frac{5}{x+6} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(y=\frac{5}{x+6}\) is \(y = 0\).

Step 1: Identifying Degree of Polynomials

The given function is \(y=\frac{5}{x+6}\). Here, the nominator is a constant (degree 0) and the denominator is a linear polynomial (degree 1). Notice that the degree of the denominator is greater than that of the numerator.

Step 2: Deciding the Horizontal Asymptote based on Degrees

If the degree of the denominator is greater than that of the numerator, the x-axis or \(y = 0\) is always the horizontal asymptote.

Step 3: Statement of the asymptote

Therefore, in our problem, the horizontal asymptote of the function is ideally \(y=0\).

Key Concepts

Rational Function

A rational function is a type of mathematical expression that can be written as a fraction of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are particularly interesting because they can have points of discontinuity and can exhibit different kinds of behavior near these points.

Key features of rational functions include:

• They can have one or more vertical asymptotes, which occur at the zeros of the denominator where the function value becomes undefined.
• The horizontal asymptote, if it exists, is determined by comparing the degree of the numerator to the degree of the denominator.

Rational functions are widely used in various fields such as physics, economics, and engineering to model real-world situations that involve rates and ratios.

Polynomials

A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. The standard form of a polynomial in one variable is \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), where \( n \) is a non-negative integer, and \( a_n \) (the leading coefficient) is not zero.

Important characteristics of polynomials include:

• They are continuous and smooth functions without any breaks or holes.
• The degree of the polynomial is determined by the highest power of the variable.
• Polynomials are easier to work with because they can be used in addition, subtraction, and multiplication, and they are defined for all input values (real numbers).

Polynomials serve as the backbone of many advanced mathematical concepts and are vital in calculus, algebra, and numerical analysis.

Degree of Polynomials

The degree of a polynomial is the highest exponent of the variable in the polynomial expression. For a polynomial \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), the degree is \( n \). It plays a crucial role in determining the behavior of polynomial functions.

Why is the degree important?

• The degree tells us about the most extreme behavior of the polynomial, as the value of \( x \) becomes very large or very small.
• In rational functions, the degree of the numerator and denominator helps determine horizontal asymptotes. As in our exercise, if the denominator's degree is greater, the horizontal asymptote is at \( y = 0 \).
• The degree also influences the number of roots (solutions) and the general shape and direction of the graph of the polynomial.

Understanding the degree of polynomials in rational functions is essential for analyzing the long-term behavior of these functions.