Problem 24 - Algebra 2

Question

Find the horizontal asymptote of the graph of each rational function. $$ y=\frac{3 x-4}{4 x+1} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(y=\frac{3x-4}{4x+1}\) is \(y = \frac{3}{4}\).

Step 1: Identify Degrees of Numerator and Denominator

Examine the degree of the numerator and the denominator. In this case, both the numerator \(3x - 4\) and the denominator \(4x + 1\) are first degree polynomials.

Step 2: Divide Coefficients

Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is obtained by dividing the coefficient of the \(x\) term in the numerator by the coefficient of the \(x\) term in the denominator.\(\frac{3}{4}\)

Step 3: Express the Horizontal Asymptote

The horizontal asymptote of this rational function can thus be described by the equation \(y = \frac{3}{4}\).

Key Concepts

Rational Functions

Rational functions are a fascinating and important type of mathematical expression used in various fields, including calculus and algebra.
A rational function is simply the ratio of two polynomials. To put it another way, it is any function that can be written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.

Rational functions are quite versatile and can be used to model real-world situations like rates and probability. Here are a few points to keep in mind:

• A rational function is only defined when the denominator \( Q(x) \) is not zero. This is because division by zero is undefined.
• The zeros of \( Q(x) \) can create vertical asymptotes, which are lines that the graph of the function approaches but never actually touches.
• The behavior of the graph at infinity (horizontal asymptotes) often relates to the degrees of the numerator and denominator.

Understanding rational functions is key to mastering more complex topics in algebra and calculus.

Degree of Polynomials

The degree of a polynomial is an important concept when studying rational functions.
It represents the highest power of the variable in the polynomial. For example, in the polynomial \( 3x^2 + 5x + 2 \), the degree is 2 because the highest power of \( x \) is 2.

This concept is crucial when determining the horizontal asymptote of a rational function:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be \( y = 0 \).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote will be a constant value, determined by the division of their leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there will be no horizontal asymptote, but there could be an oblique asymptote.

Understanding the degree of polynomials helps explain the behavior of rational functions as they approach infinity.

Coefficients of Polynomials

Coefficients in polynomials play a pivotal role, especially when dealing with rational functions. They are essentially the numbers in front of the variables in a polynomial.

For example, in the expression \( 3x^2 + 2x + 1 \), 3, 2, and 1 are the coefficients.

• The leading coefficient is the coefficient of the highest degree term in the polynomial. In \( 4x^3 + x^2 -6 \), 4 is the leading coefficient.
• These coefficients are crucial when finding the horizontal asymptote for a rational function, particularly when the degrees of the numerator and denominator are equal.
• To calculate the horizontal asymptote, you divide the leading coefficient of the numerator by the leading coefficient of the denominator for rational functions of equal degree.

This concept is not only important for solving equations but also in understanding the graphical behavior of different polynomials.