Find the Asymptotes (2x^4+6x^3-36x^2)/(x^4-45x^2+324)
The question is asking for the identification of the asymptotes of the given rational function. Asymptotes are lines that the graph of the function approaches but never actually reaches. There are different types of asymptotes - vertical, horizontal, and oblique (or slant). In the context of this function, which is a ratio of two polynomials, vertical asymptotes occur where the denominator equals zero (and the numerator is not also zero), potentially indicating places where the function is undefined. Horizontal asymptotes are concerned with the behavior of the function as the independent variable (often 'x') goes to positive or negative infinity, often determined by the degrees and leading coefficients of the polynomials in the numerator and denominator. An oblique asymptote may exist if the degree of the polynomial in the numerator is exactly one greater than that of the denominator. The question requires analyzing the function to determine if any of these asymptotes are present and, if so, to describe them.
$\frac{2 x^{4} + 6 x^{3} - 36 x^{2}}{x^{4} - 45 x^{2} + 324}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $\frac{2x^4 + 6x^3 - 36x^2}{x^4 - 45x^2 + 324}$ is not defined. These values are $x = -6, -3, 3, 6$.
Step 2:
Analyze the behavior of the function as $x$ approaches $-3$. As $x \to -3^-$, the function tends to $\infty$, and as $x \to -3^+$, it tends to $-\infty$. Thus, $x = -3$ is a vertical asymptote.
Step 3:
Examine the behavior of the function as $x$ approaches $6$. As $x \to 6^-$, the function tends to $-\infty$, and as $x \to 6^+$, it tends to $\infty$. Hence, $x = 6$ is a vertical asymptote.
Step 4:
Compile a list of all vertical asymptotes: $x = -3, 6$.
Step 5:
Review the rules for horizontal asymptotes for a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator:
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 $n$ and $m$ of the numerator and denominator, respectively. Here, $n = 4$ and $m = 4$.
Step 7:
Since $n = m$, the horizontal asymptote is given by $y = \frac{a}{b}$, where $a = 2$ and $b = 1$. Therefore, $y = 2$ is the horizontal asymptote.
Step 8:
Conclude that there are no oblique asymptotes since the degree of the numerator is equal to the degree of the denominator.
Step 9:
Summarize all asymptotes:
- Vertical Asymptotes: $x = -3, 6$
- Horizontal Asymptotes: $y = 2$
- No Oblique Asymptotes
Step 10:
End of the problem-solving process.
Knowledge Notes:
Vertical Asymptotes: These occur at values of $x$ where the denominator of a rational function is zero, and the numerator is not zero. The function tends to $\pm\infty$ as $x$ approaches these values.
Horizontal Asymptotes: These occur when the degrees of the numerator and denominator are the same, or the degree of the numerator is less than the degree of the denominator. The horizontal asymptote is $y = \frac{a}{b}$ if the degrees are equal, and $y = 0$ if the degree of the numerator is less.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the rational function may have an oblique (slant) asymptote, which is found by polynomial long division.
Rational Functions: A function of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
Behavior at Infinity: To determine the horizontal asymptote, if any, one can also look at the leading coefficients and degrees of the polynomials in the numerator and denominator as $x$ approaches $\pm\infty$.
Undefined Points: The points where the function is not defined (denominator equals zero) are critical in finding vertical asymptotes, but they are not always asymptotes if they cancel out with factors in the numerator.
