Find the Asymptotes (3x+9)/(x^2-2x-3)

Solution:

Step 1:

Determine the values of $x$ for which the function $f(x) = \frac{3x + 9}{x^2 - 2x - 3}$ is not defined, which occur where the denominator equals zero. Solve $x^2 - 2x - 3 = 0$ to find $x = -1$ and $x = 3$.

Step 2:

Analyze the behavior of $f(x)$ as $x$ approaches $-1$. As $x$ approaches $-1$ from the left, $f(x)$ goes to positive infinity, and from the right, it goes to negative infinity. Therefore, $x = -1$ is a vertical asymptote.

Step 3:

Examine the behavior of $f(x)$ as $x$ approaches $3$. As $x$ approaches $3$ from the left, $f(x)$ goes to negative infinity, and from the right, it goes to positive infinity. Hence, $x = 3$ is a vertical asymptote.

Step 4:

Compile a list of all vertical asymptotes found, which are $x = -1$ and $x = 3$.

Step 5:

To find horizontal asymptotes for a rational function $R(x) = \frac{ax^n}{bx^m}$, compare the degrees of the numerator ($n$) and denominator ($m$). If $n < m$, the horizontal asymptote is $y = 0$. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$. If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.

Step 6:

Identify the degrees of the numerator ($n$) and denominator ($m$) for our function. Here, $n = 1$ and $m = 2$.

Step 7:

Since the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is the x-axis, given by $y = 0$.

Step 8:

An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.

Step 9:

Summarize the asymptotes of the function: Vertical Asymptotes are at $x = -1$ and $x = 3$, the Horizontal Asymptote is $y = 0$, and there are No Oblique Asymptotes.

Knowledge Notes:

To find the asymptotes of a rational function, one must analyze the function's behavior as the independent variable approaches specific critical values. Here are the relevant knowledge points:

1. Vertical Asymptotes: These occur at values of $x$ where the denominator of a rational function is zero (provided the numerator is not also zero at these points), because the function values approach infinity or negative infinity.

2. Horizontal Asymptotes: These are found by comparing the degrees of the numerator and denominator. If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y=0$. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If $n > m$, there is no horizontal asymptote.

3. Oblique Asymptotes: These may occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, one would perform polynomial long division to find the equation of the oblique asymptote.

4. Behavior at Asymptotes: To determine the nature of a vertical asymptote, one should check the limits of the function as $x$ approaches the asymptote from the left and from the right. If the limits are infinite but have different signs, the vertical line at that $x$ value is a vertical asymptote.

5. Solving Quadratic Equations: To find the values where the function is undefined, one may need to solve a quadratic equation, which can be done by factoring, using the quadratic formula, or completing the square.

Understanding these concepts allows one to accurately determine the asymptotic behavior of rational functions.