Problem 22 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). \(s(x)=\frac{3 x^{2}}{x^{2}+2 x+5}\)

Step-by-Step Solution

Answer

Horizontal asymptote: \(y=3\). No vertical asymptotes.

Step 1: Check for Horizontal Asymptote

To find the horizontal asymptote, we need to compare the degrees of the polynomial in the numerator and the polynomial in the denominator. In the function \( s(x) = \frac{3x^2}{x^2 + 2x + 5} \), both the numerator and the denominator are of degree 2. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of these polynomials. Here, the leading coefficients are 3 (numerator) and 1 (denominator). Thus, the horizontal asymptote is \( y = \frac{3}{1} = 3 \).

Step 2: Check for Vertical Asymptotes

To find vertical asymptotes, we need to set the denominator equal to zero and solve for \( x \). The denominator is \( x^2 + 2x + 5 \). We solve the equation \( x^2 + 2x + 5 = 0 \) by finding the roots. Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = 5 \), we have: \( x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} \). Here, the discriminant is negative, so the quadratic has no real roots, indicating there are no vertical asymptotes.

Key Concepts

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph of a function approaches as the input, or x-value, goes to infinity or negative infinity. It tells us about the end behavior of a function. To find a horizontal asymptote in a rational function like\[s(x) = \frac{3x^2}{x^2 + 2x + 5},\]we need to compare the degrees of the numerator and the denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line \(y = 0\).
• If the degrees are equal, as in our case, the horizontal asymptote is the ratio of their leading coefficients. Here, both polynomials are of degree 2 (the highest power is 2), and the numerator's leading coefficient is 3 while the denominator's leading coefficient is 1. Therefore, the horizontal asymptote is \(y = \frac{3}{1} = 3\).
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Understanding horizontal asymptotes helps us predict how the function behaves when the x-values get extremely large or small. The function tends to get closer to the line \(y = 3\) but never actually reaches it.

Vertical Asymptote

Vertical asymptotes occur where the denominator of a rational function equals zero, provided that the numerator is not zero at these points. These are places where the function heads towards positive or negative infinity. Here's how we find them using the given function:\[s(x) = \frac{3x^2}{x^2 + 2x + 5}.\]To check for vertical asymptotes, set the denominator\[x^2 + 2x + 5\]equal to zero and solve for \(x\). This requires us to solve:\[x^2 + 2x + 5 = 0.\]In this case, using the quadratic formula\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]with \(a = 1\), \(b = 2\), and \(c = 5\), leads us to:\[x = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2}.\]The discriminant \(4 - 20\) is negative, meaning there are no real solutions, and therefore, no vertical asymptotes for this function. If the roots were real, they would indicate the x-values where these vertical asymptotes are located.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula is\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]It allows us to find the roots of the equation, which are the values of \(x\) that make the equation equal to zero.
Here's how it works:

• The symbol \(b^2 - 4ac\) is called the discriminant. Its value determines the nature of the roots of the equation.
• If the discriminant is positive, there are two distinct real roots.
• If it's zero, there is exactly one real root.
• If it's negative, the roots are complex and there are no real solutions.

In our original exercise, the quadratic formula was used to find the roots of \(x^2 + 2x + 5 = 0\). Given that the discriminant was negative, it meant there were no real roots, suggesting no vertical asymptotes for the given rational function. Understanding these algebraic tools provides insight into the behavior of functions and is essential for tackling various types of mathematical problems.