Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. $$f(x)=\frac{3-14 x-5 x^{2}}{3+7 x+2 x^{2}}$$

Step-by-Step Solution

Answer

The rational function \(f(x)=\frac{3-14x-5x^{2}}{3+7x+2x^{2}}\) has no holes, vertical asymptotes at \(x=-1\) and \(x=-1.5\), and horizontal asymptote at \(y=-5/2\).

Step 1: Factorize the numerator and the denominator

First, both the numerator and the denominator of the rational function \(f(x)=\frac{3-14x-5x^{2}}{3+7x+2x^{2}}\) are polynomial functions, which should be factored completely. However, neither the numerator nor the denominator can be factored any further.

Step 2: Identify any holes

A hole in a rational function occurs when a term in the numerator and a term in the denominator cancel each other out. Because there are no terms that cancel out in this function, we can conclude that the function has no holes.

Step 3: Identify any vertical asymptotes

A vertical asymptote of a rational function occurs when the denominator equals zero. So, we need to solve the equation \(3+7x+2x^{2}=0\). Using the quadratic formula, we obtain two roots, \(x=-1\) and \(x=-1.5\), which are the vertical asymptotes of the function.

Step 4: Identify any horizontal asymptotes

To identify a horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator. Both have a degree of 2, and thus by looking at the coefficients of the highest-degree terms, we determine that there is a horizontal asymptote at \(y=-5/2\).

Step 5: Verify your answers using a graphing utility

Finally, use a graphing utility to graph the function and verify that the function approaches but never touches the lines \(x=-1\), \(x=-1.5\), and \(y=-5/2\), and that there are no holes in the graph.

Key Concepts

Asymptotes

Understanding asymptotes is crucial when graphing rational functions. Asymptotes are lines that the graph of a function approaches but never reaches. There are two main types of asymptotes: vertical and horizontal.

Vertical asymptotes occur when the function's denominator equals zero and it is not canceled out by the numerator. To find them, set the denominator equal to zero and solve for the variable. In our exercise, the denominator is given as \(3+7x+2x^2\) and by solving \(3+7x+2x^2=0\), we find the vertical asymptotes at \(x=-1\) and \(x=-1.5\).

Horizontal asymptotes are found by comparing degrees of the numerator and denominator. If the degrees are the same, the horizontal asymptote will be the division of the leading coefficients. In this case, both polynomials are of degree 2, leading to a horizontal asymptote at \(y=-5/2\).

Holes in Rational Functions

Holes are points on the graph where the function is not defined, which occur when a factor in the numerator and the denominator of the rational function cancel each other out. To check for holes, factor both the numerator and denominator and simplify.

For instance, if a rational function had a factor of \(x-a\) in both the top and bottom, it would have a hole at \(x=a\). However, since the function \(f(x)=\frac{3-14x-5x^{2}}{3+7x+2x^{2}}\) cannot be further factored and there are no common factors, it confirms that the function has no holes.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It involves breaking down a polynomial into a product of simpler polynomials or factors that, when multiplied together, give back the original polynomial. Common factoring techniques include finding a common factor, using the difference of squares, and factoring trinomials.

In this exercise, one might have tried to find factors of the numerator and denominator by looking for common factors or by employing the quadratic formula if the polynomials were factorable trinomials. However, since neither \(3-14x-5x^2\) nor \(3+7x+2x^2\) could be factored further, we proceed without factoring.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2+bx+c=0\). The formula is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the polynomial.

In the context of rational functions, this formula is often used to find vertical asymptotes by solving for when the denominator equals zero. In our given function, by applying the quadratic formula to the denominator, we find the roots \(x=-1\) and \(x=-1.5\), which indicate the vertical asymptotes.