Problem 25 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). $$ s(x)=\frac{6 x^{2}+1}{2 x^{2}+x-1} $$

Step-by-Step Solution

Answer

The horizontal asymptote is \( y = 3 \); vertical asymptotes are \( x = \frac{1}{2} \) and \( x = -1 \).

Step 1: Determine Horizontal Asymptotes

To find horizontal asymptotes, compare the degrees of the polynomial in the numerator and the polynomial in the denominator. Both numerator and denominator are degree 2 polynomials. For rational functions where the degrees are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. The leading coefficient of the numerator is 6, and the leading coefficient of the denominator is 2, which gives:\[y = \frac{6}{2} = 3\]Thus, the horizontal asymptote is \( y = 3 \).

Step 2: Determine Vertical Asymptotes

Finding vertical asymptotes requires us to set the denominator equal to zero and solve for \( x \). Solve for \( x \) using the equation:\[ 2x^2 + x - 1 = 0 \]Factor the quadratic:\( 2x^2 + x - 1 = 0 \)Factoring, \((2x - 1)(x + 1) = 0 \)Set each factor to zero and solve for \( x \):\[2x - 1 = 0 \rightarrow x = \frac{1}{2} \\]\[x + 1 = 0 \rightarrow x = -1 \\]Therefore, the vertical asymptotes are \( x = \frac{1}{2} \) and \( x = -1 \).

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. It's important to note that the denominator of a rational function can't be zero since division by zero is undefined. This type of function is expressed in the form:

• \( f(x) = \frac{P(x)}{Q(x)} \)

where \( P(x) \) and \( Q(x) \) are polynomials. The properties of rational functions are intriguing, especially when investigating asymptotes, which are lines that the graph approaches as it extends towards infinity. The two main types of asymptotes we'll discuss are horizontal and vertical asymptotes.

Polynomial Degree Comparison

When working with rational functions, understanding the degree of the polynomials involved is crucial. The degree of a polynomial is the highest power of the variable in the expression. In the given problem, the function:

• \( s(x)=\frac{6x^{2}+1}{2x^{2}+x-1} \)

has a numerator and a denominator, each of degree 2. This implies that they have the same degree. Comparing the degrees of the numerator and the denominator helps identify the type and location of horizontal asymptotes.
When both polynomial degrees are equal, the horizontal asymptote can be determined from the leading coefficients.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. Identifying the leading coefficients is essential in finding the horizontal asymptotes of a rational function. In our example, the leading coefficients are:

• Numerator (6)
• Denominator (2)

To find the horizontal asymptote for rational functions where the degrees are equal, calculate the ratio of the leading coefficients:
\( y = \frac{6}{2} = 3 \).
Thus, the horizontal asymptote for the given function is \( y = 3 \). Understanding and using leading coefficients makes complex polynomial relationships approachable.

Factoring Quadratics

Factoring is a method used to simplify quadratic expressions, usually aiming to find the roots or solve equations. It is particularly useful for determining vertical asymptotes in rational functions. To find the vertical asymptotes, we focus on the denominator and set it to zero:

• \( 2x^2 + x - 1 = 0 \)

Factoring this quadratic, we express it as:

• \((2x - 1)(x + 1) = 0\)

We solve for each factor to find where the denominator becomes zero:
\( 2x - 1 = 0 \) gives \( x = \frac{1}{2} \), and \( x + 1 = 0 \) gives \( x = -1 \).
So, the vertical asymptotes are \( x = \frac{1}{2} \) and \( x = -1 \). Factoring quadratics effectively opens up the underlying math, allowing us to explore the behavior of rational functions around roots.