Find the Asymptotes (4x^4+12x^3-72x^2)/(x^4-45x^2+324)
The question is asking to determine the lines that the graph of the given rational function (4x^4 + 12x^3 - 72x^2) / (x^4 - 45x^2 + 324) approaches as the x-values head towards positive or negative infinity (horizontal asymptotes), or the lines that the graph approaches as the x-values head towards specific values that are not in the function's domain (vertical asymptotes). Asymptotes can be thought of as the behavior boundaries of the graph of a function, where the function gets infinitely close but never actually touches the asymptote. The problem requires an understanding of how to analyze rational functions and find their asymptotic behavior.
$\frac{4 x^{4} + 12 x^{3} - 72 x^{2}}{x^{4} - 45 x^{2} + 324}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $\frac{4x^4 + 12x^3 - 72x^2}{x^4 - 45x^2 + 324}$ does not exist. These values are $x = -6, -3, 3, 6$.
Step 2:
Analyze the behavior of the function as $x$ approaches $-3$. The function tends to positive infinity from the left and negative infinity from the right, indicating a vertical asymptote at $x = -3$.
Step 3:
Examine the function's behavior as $x$ approaches $6$. The function approaches negative infinity from the left and positive infinity from the right, confirming $x = 6$ as another vertical asymptote.
Step 4:
Compile the list of vertical asymptotes: $x = -3, 6$.
Step 5:
Review the conditions for horizontal asymptotes in a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ and $m$ are the degrees of the numerator and denominator, respectively.
Step 6:
Identify the degrees $n$ and $m$ for the given function. Here, $n = 4$ and $m = 4$.
Step 7:
Since $n$ equals $m$, the horizontal asymptote is given by the line $y = \frac{a}{b}$, where $a = 4$ and $b = 1$. Thus, the horizontal asymptote is $y = 4$.
Step 8:
Conclude that there are no oblique asymptotes, as the degree of the numerator is not greater than that of the denominator.
Step 9:
Summarize all asymptotes found for the function:
- Vertical Asymptotes: $x = -3, 6$
- Horizontal Asymptote: $y = 4$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, we follow these steps:
Vertical Asymptotes: These occur at the values of $x$ that make the denominator zero, as long as those values do not also make the numerator zero (which would indicate a hole instead). To find these, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator ($n$) and denominator ($m$):
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
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.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, perform polynomial long division and take the quotient (ignoring the remainder).
Behavior Around Vertical Asymptotes: To determine the direction of the approach to vertical asymptotes, evaluate the limit of the function as $x$ approaches the asymptote from both the left and the right.
Combining Asymptotes: The set of all asymptotes provides a sketch of the behavior of the function at infinity and near critical points where the function is undefined.
