Of course! Let's analyze the expression [tex]\( y = \frac{2x^2}{x^2 - 5x - 6} \)[/tex] step-by-step, decompose it into partial fractions, and identify the asymptotes.
### Step 1: Factor the Denominator
First, we need to factor the quadratic expression in the denominator:
[tex]\[ x^2 - 5x - 6 \][/tex]
This factors as:
[tex]\[ (x - 6)(x + 1) \][/tex]
### Step 2: Decompose into Partial Fractions
Next, we decompose the expression into partial fractions. This involves rewriting the expression in the form:
[tex]\[ \frac{2x^2}{(x - 6)(x + 1)} \][/tex]
as:
[tex]\[ \frac{A}{x - 6} + \frac{B}{x + 1} + C \][/tex]
By comparing coefficients or using partial fraction decomposition techniques, we find that:
[tex]\[ y = 2 - \frac{2}{7(x + 1)} + \frac{72}{7(x - 6)} \][/tex]
Therefore,
[tex]\[ \frac{2x^2}{(x - 6)(x + 1)} = 2 - \frac{2}{7(x + 1)} + \frac{72}{7(x - 6)} \][/tex]
### Step 3: Identify Vertical Asymptotes
Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at those points. The factors of the denominator give us these points, specifically:
[tex]\[ x - 6 = 0 \implies x = 6 \][/tex]
[tex]\[ x + 1 = 0 \implies x = -1 \][/tex]
Thus, the vertical asymptotes are at:
[tex]\[ x = -1 \][/tex]
[tex]\[ x = 6 \][/tex]
### Step 4: Determine the Horizontal Asymptote
To find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Both the numerator ([tex]\(2x^2\)[/tex]) and the denominator ([tex]\(x^2 - 5x - 6\)[/tex]) are of degree 2. When the degrees of the numerator and the denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients.
Here, the leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is:
[tex]\[ y = \frac{2}{1} = 2 \][/tex]
### Summary
- Partial Fraction Decomposition:
[tex]\[ \frac{2x^2}{(x - 6)(x + 1)} = 2 - \frac{2}{7(x + 1)} + \frac{72}{7(x - 6)} \][/tex]
- Vertical Asymptotes:
[tex]\[ x = -1 \][/tex]
[tex]\[ x = 6 \][/tex]
- Horizontal Asymptote:
[tex]\[ y = 2 \][/tex]
This concludes our detailed step-by-step solution for the problem.
