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Horizontal asymptotes are an important concept in understanding the behavior of rational functions as they approach infinity. They help us visualize what happens to the value of a function when the input grows extremely large or small. To find horizontal asymptotes, we compare the degrees of the polynomial in the numerator and the denominator.

When the degree of the numerator is equal to the degree of the denominator, as in the function \(s(x)=\frac{3x^2}{x^2+2x+5}\), the horizontal asymptote is simply the ratio of the leading coefficients. Here, both the numerator and the denominator have a degree of 2. This means that the horizontal asymptote is \(y=\frac{3}{1}=3\).

Horizontal asymptotes indicate that as \(x\) approaches infinity, the function \(s(x)\) gets closer and closer to the value 3.

Vertical asymptotes occur where a function is undefined due to a division by zero in the formula. To find these, we set the denominator of the function equal to zero and solve for \(x\). For \(s(x)=\frac{3x^2}{x^2+2x+5}\), we solve the equation \(x^2 + 2x + 5 = 0\).

However, solving for \(x\), we encounter a concept known as the discriminant. The discriminant, obtained from the quadratic formula, helps determine the nature of the roots without finding their exact values. In this case, \(b^2 - 4ac = 2^2 - 4 imes 1 imes 5 = -16\).

A negative discriminant indicates that there are no real roots, meaning the denominator never becomes zero for real numbers. Therefore, this function has no vertical asymptotes, as it remains finite for all real \(x\).

The discriminant is a key component in algebra used to determine the number and nature of the roots of a quadratic equation. It is derived from the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(b^2 - 4ac\) is the discriminant.

Here's how it works:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, indicating the roots are repeated.
• If the discriminant is negative, the quadratic equation has no real roots—only complex or imaginary roots.

In the function \(s(x)\), solving \(x^2 + 2x + 5 = 0\) gives us a discriminant of \(-16\). Since this is negative, it confirms that there are no points where the denominator will equal zero, aligning with there being no vertical asymptotes for the function.

Understanding the discriminant not only helps in identifying vertical asymptotes but also gives insight into the overall behavior of quadratic expressions.