Problem 32 - College Algebra

Question

Find the horizontal asymptote, if any, of the graph of each rational function. $$g(x)=\frac{15 x^{2}}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(g(x) = \frac{15x^{2}}{3x^{2} + 1}\) is \(y = 5\).

Step 1: Determine the degree of the numerator and the denominator

The degree of a polynomial is the highest power of x in its term. The degree of the numerator \(15x^{2}\) is 2 and the degree of the denominator \(3x^{2} + 1\) is also 2 as the highest power of x in the denominator is 2.

Step 2: Compare the degrees of the numerator and the denominator

Since the degrees of the numerator and the denominator are equal, we can find the horizontal asymptote by taking the ratio of the leading coefficients of the numerator and the denominator.

Step 3: Compute the horizontal asymptote

The leading coefficient of the numerator is 15 and of the denominator is 3. So, the horizontal asymptote is given by \(\frac{15}{3}\), which simplifies to 5.

Key Concepts

Rational Function

When studying graphs of functions, one type might catch your attention due to its interesting behavior as the input values grow large. This type is known as a rational function. A rational function is a fraction where both the numerator and the denominator are polynomials. The function g(x) = \(\frac{15 x^{2}}{3 x^{2}+1}\) from our exercise is an example of this.

In essence, rational functions can have curves that approach a specific value, but never actually reach that value. These are called asymptotes. A common type is a horizontal asymptote, which tells us the value that the y-coordinate of the function approaches as x tends toward positive or negative infinity. Understanding these can provide great insights into how the function behaves at extreme values of x. And interestingly, even slight changes in the form of a rational function can lead to big differences in its graph!

Degree of a Polynomial

Every polynomial has a characteristic called its degree, which is a crucial concept in algebra. The degree of a polynomial is determined by the highest power of the variable within the polynomial. For instance, in g(x) = \(\frac{15 x^{2}}{3 x^{2}+1}\), the degree is 2, since the highest exponent on the variable x is 2.

Why is this important? The degree of a polynomial tells us a lot about the function's behavior, especially its end behavior as x becomes very large or very small. For rational functions, the degrees of the numerator and denominator polynomials influence whether horizontal asymptotes exist and what they are. As a quick rule of thumb, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If they are equal, like in our case, the horizontal asymptote is found by the ratio of the leading coefficients.

Leading Coefficient

Within the tapestry of polynomials, each term has its own coefficient. The leading coefficient is the Nostradamus of the group – it is the coefficient of the term with the highest degree, which takes the lead in forecasting the function's long-term trends.

In the function g(x) = \(\frac{15 x^{2}}{3 x^{2}+1}\), the leading coefficient of the numerator is 15 and that of the denominator is 3. And just as a captain steers a ship, the leading coefficients guide us to the value of the horizontal asymptote for rational functions when the degrees of the numerator and denominator are equal. By comparing these two valiant leaders - dividing 15 by 3 - we confidently sail towards our horizontal asymptote: the value y=5.

Remember, while the leading coefficient might seem like just another number, its role is pivotal. It not only influences the slope of lines in linear equations but also determines the width of the parabola in quadratic equations, and more broadly, the end behavior of higher degree polynomials.