Problem 69 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x) .\) You may want to use division, factoring, or transformations as an aid. Show all asymptotes and "holes." $$ f(x)=\frac{2 x^{2}+9 x+9}{2 x^{2}+7 x+6} $$

Step-by-Step Solution

Answer

The function has vertical asymptotes at \(x = -1.5\) and \(x = -2\), and a horizontal asymptote at \(y = 1\).

Step 1: Identify Domain

First, determine the domain of the function by finding the values for which the denominator is zero, since division by zero is undefined. Set the denominator equal to zero and solve for \(x\): \(2x^2 + 7x + 6 = 0\).

Step 2: Solve Denominator Equation

Solve \(2x^2 + 7x + 6 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a = 2\), \(b = 7\), \(c = 6\). Calculate the discriminant: \(b^2 - 4ac = 49 - 48 = 1\). Thus, the roots are \(x = \frac{-7 \pm 1}{4}\) resulting in \(x = -1.5\) and \(x = -2\).

Step 3: Determine Asymptotes and Holes

Since the denominator has two roots \(x = -1.5\) and \(x = -2\), check if the numerator \(2x^2 + 9x + 9\) has these roots too to determine if they are holes: Substitute these x-values into the numerator. Both do not zero the numerator. Hence, \(x = -1.5\) and \(x = -2\) are vertical asymptotes.

Step 4: Simplify the Function

Factor the numerator \(2x^2 + 9x + 9\) and check if there is any cancellation option with the denominator. It factors as \((2x + 3)(x + 3)\). The denominator factors as \((2x + 3)(x + 2)\). Simplifying gives \(f(x) = \frac{(x+3)}{(x+2)}\).

Step 5: Identify Horizontal Asymptote

A rational function has a horizontal asymptote at \(y = \frac{a}{b}\) if the degrees of the numerator and denominator polynomials are equal. Here, both are degree 1 after simplification, so the horizontal asymptote is \(y = \frac{1}{1} = 1\).

Step 6: Plot and Analyze the Graph

Sketch the graph including the asymptotes and behavior analysis at \(x = -1.5\), \(x = -2\), and horizontal asymptote at \(y = 1\). The graph approaches but never touches these lines.

Key Concepts

Vertical asymptotes

In the world of rational functions, vertical asymptotes are invisible lines that the graph of the function approaches but never actually touches or crosses. These occur at the values of \(x\) where the denominator of the function becomes zero, leading to division by zero, which is undefined. To find the vertical asymptotes in a rational function like \(f(x) = \frac{2x^2 + 9x + 9}{2x^2 + 7x + 6}\), we first set the denominator equal to zero. Solving \(2x^2 + 7x + 6 = 0\) using techniques such as factoring or the quadratic formula reveals these critical \(x\)-values. After solving, we identified \(x = -1.5\) and \(x = -2\) as vertical asymptotes since neither of these values zeroed the numerator. This means that at these points, the graph will shoot up to positive or negative infinity, illustrating the behavior typical of vertical asymptotes.

Horizontal asymptotes

Horizontal asymptotes reflect the end-behavior of a rational function, indicating the value that the function approaches as \(x\) becomes extremely large or small. To determine horizontal asymptotes, we compare the degrees of the numerator and the denominator. In general, we consider three cases:

• If the degree of the numerator is less than that of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and the denominator, respectively.
• If the degree of the numerator is greater, there is no horizontal asymptote.

For the function \(f(x) = \frac{2x^2 + 9x + 9}{2x^2 + 7x + 6}\), after simplifying, both numerator and denominator have degree 1, and their leading coefficients are both 1, hence the horizontal asymptote is \(y = 1\). This helps in understanding the overall trend of the graph as \(x\) becomes very large or very small.

Quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula gives us the roots of a quadratic equation, where \(a\), \(b\), and \(c\) are constants. The term under the square root, \(b^2 - 4ac\), is called the discriminant. It provides insights into the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is one real root (a repeated root).
• If it is negative, the roots are complex and not real numbers.

In the context of rational functions, solving the denominator using the quadratic formula not only helps in finding vertical asymptotes but also checking for potential holes if the numerator shares the same roots. For instance, in our given function, solving the denominator \(2x^2 + 7x + 6 = 0\) with the quadratic formula provided the roots which clarified the vertical asymptotes.