Problem 28 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{6 x^{2}-x-2}{2 x^{2}+x-6} $$

Step-by-Step Solution

Answer

Vertical asymptotes are at \(x = \frac{3}{2}\) and \(x = -2\). The horizontal asymptote is \(y = 3\).

Step 1: Identify the Rational Function Form

The function given is \( f(x) = \frac{6x^2 - x - 2}{2x^2 + x - 6} \). It is in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) = 6x^2 - x - 2 \) and \( Q(x) = 2x^2 + x - 6 \). We'll use this form to find the asymptotes.

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator \( Q(x) \) equals zero, but the numerator \( P(x) \) does not equal zero. Set \( Q(x) = 0 \) and solve: \[ 2x^2 + x - 6 = 0. \] Factoring gives \( (2x - 3)(x + 2) = 0 \). Thus, \( x = \frac{3}{2} \) and \( x = -2 \). Check that these do not cancel with any factor from \( P(x) \). Both \( x = \frac{3}{2} \) and \( x = -2 \) are vertical asymptotes.

Step 3: Determine Horizontal Asymptote

Horizontal asymptotes depend on the degrees of \( P(x) \) and \( Q(x) \). Here, both \( P(x) \) and \( Q(x) \) are degree 2. Hence, the horizontal asymptote is \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \), respectively. Thus, the horizontal asymptote is \( y = \frac{6}{2} = 3 \).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where a function grows infinitely large in the positive or negative direction. These occur in rational functions when the denominator equals zero, but the numerator does not. For the function \( f(x) = \frac{6x^2 - x - 2}{2x^2 + x - 6} \), vertical asymptotes can be found by solving \( Q(x) = 0 \).
Solving the equation \( 2x^2 + x - 6 = 0 \) by factoring, we get \((2x - 3)(x + 2) = 0\).
This results in the solutions \( x = \frac{3}{2} \) and \( x = -2 \).
Since neither of these terms cancel with factors from the numerator, each value is a vertical asymptote.

• Vertical asymptotes create a boundary in the graph where the function is undefined.
• Graphically, the lines \( x = \frac{3}{2} \) and \( x = -2 \) divide the function's behavior into different segments.

Understanding vertical asymptotes helps in analyzing how the function behaves near these undefined regions.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches positive or negative infinity. They are key in understanding the long-term behavior of rational functions like \( f(x) = \frac{6x^2 - x - 2}{2x^2 + x - 6} \).
To determine a horizontal asymptote, consider the degrees of the polynomial expressions in the numerator and denominator.
If the degrees are equal, as with this function, the horizontal asymptote is the ratio of the leading coefficients.
In this example, both the numerator and denominator are degree 2, with leading coefficients 6 and 2, respectively.
Thus, the horizontal asymptote is \( y = \frac{6}{2} = 3 \).

• This means that as \( x \) becomes very large, the function will approximate \( y = 3 \).
• Horizontal asymptotes can help predict the end behavior of graphs effectively.

This insight helps students understand that the function eventually stabilizes at this value.

Polynomial Degrees

The degree of a polynomial is the highest power of its variable. In rational functions, the degree influences the function’s asymptotic behavior. For the function \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) = 6x^2 - x - 2 \) and \( Q(x) = 2x^2 + x - 6 \), each is a degree 2 polynomial.
The relationship between the degrees of the numerator and the denominator informs the location of the horizontal asymptote.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are the same, the horizontal asymptote is the ratio of leading coefficients.
• When the numerator's degree exceeds the denominator's, there is no horizontal asymptote, but there might be an oblique asymptote.

This concept is crucial because it allows one to quickly assess the rational function’s asymptotic nature without detailed calculations. Understanding polynomial degrees gives clarity and depth to function analysis and graph interpretation.