Find the Asymptotes (x^2)/(x^2-36)
The question is asking you to determine the lines that the graph of the given rational function approaches as the x-values head towards infinity or negative infinity, or as the x-values approach specific points where the function is undefined. These lines are called asymptotes, and they can be vertical, horizontal, or oblique (slanted). The function provided is (x^2)/(x^2-36), and you would typically look for vertical asymptotes by finding the values of x that make the denominator zero (since the function is undefined at these points) and horizontal or oblique asymptotes by analyzing the behavior of the function as x approaches infinity or negative infinity.
$\frac{x^{2}}{x^{2} - 36}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $f(x) = \frac{x^2}{x^2 - 36}$ is not defined. These occur when the denominator equals zero: $x^2 - 36 = 0$, which gives $x = \pm 6$.
Step 2:
Analyze the behavior of $f(x)$ as $x$ approaches $-6$. As $x \to -6^-$, $f(x) \to \infty$, and as $x \to -6^+$, $f(x) \to -\infty$. Therefore, $x = -6$ is a vertical asymptote.
Step 3:
Examine the behavior of $f(x)$ as $x$ approaches $6$. As $x \to 6^-$, $f(x) \to -\infty$, and as $x \to 6^+$, $f(x) \to \infty$. Hence, $x = 6$ is also a vertical asymptote.
Step 4:
Compile a list of the vertical asymptotes identified: $x = -6$ and $x = 6$.
Step 5:
Consider the general form of a rational function $R(x) = \frac{a x^n}{b x^m}$, where $n$ and $m$ are the degrees of the numerator and denominator, respectively. The horizontal asymptote depends on the relationship between $n$ and $m$:
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 (an oblique asymptote may exist).
Step 6:
Identify the degrees $n$ and $m$ for the given function. Here, $n = 2$ and $m = 2$.
Step 7:
Since $n$ equals $m$, the horizontal asymptote is given by $y = \frac{a}{b}$. For our function, $a = 1$ and $b = 1$, so the horizontal asymptote is $y = 1$.
Step 8:
Determine the presence of any oblique asymptotes. Since the degree of the numerator is not greater than the degree of the denominator, there are no oblique asymptotes.
Step 9:
Summarize the asymptotes of the function:
- Vertical Asymptotes: $x = -6$ and $x = 6$
- Horizontal Asymptote: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
A vertical asymptote occurs at values of $x$ where the function becomes unbounded (approaches $\infty$ or $-\infty$) and is typically found where the denominator of a rational function is zero.
A horizontal asymptote describes the behavior of a function as $x$ approaches $\infty$ or $-\infty$. It is determined by comparing the degrees of the numerator and denominator of the rational function:
If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y = 0$.
If $n$ equals $m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
If $n$ is greater than $m$, there is no horizontal asymptote, but there may be an oblique asymptote if the difference in degrees is one.
An oblique asymptote (or slant asymptote) occurs when the degree of the numerator is exactly one more than the degree of the denominator. It can be found by performing polynomial long division.
To find asymptotes, one must analyze the function's behavior at critical points and as $x$ approaches infinity or negative infinity.
