Find the Asymptotes (-x^2+5x-4)/(5x^2+5x-10)
The question is asking for the asymptotes of the rational function provided. Asymptotes are lines that the graph of a function approaches as the inputs or outputs become large. There are two types of asymptotes that are typically of interest for this type of function: horizontal and vertical asymptotes.
Vertical asymptotes occur where the function is undefined due to division by zero. That is, they are found by determining the values of x for which the denominator of the rational function equals zero, as long as these values do not also make the numerator zero (in which case they might be holes instead).
Horizontal asymptotes are about the behavior of the function as x goes to positive or negative infinity. These are found by comparing the degrees of the numerator and the denominator and considering the coefficients for the highest power of x.
The question does not ask about oblique (slant) asymptotes, which occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, you would divide the numerator by the denominator and the oblique asymptote would be the linear part of the quotient.
Remember, the question does not want an actual calculation or result for the asymptotes, but is inquiring about the process to find them for the given rational function.
$\frac{- x^{2} + 5 x - 4}{5 x^{2} + 5 x - 10}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the function $f(x) = \frac{-x^2 + 5x - 4}{5x^2 + 5x - 10}$ to be undefined. These are the roots of the denominator: $x = -2, x = 1$.
Step 2:
Examine the behavior of $f(x)$ as $x$ approaches $-2$. As $x \to -2^-$, $f(x) \to -\infty$, and as $x \to -2^+$, $f(x) \to \infty$. Thus, $x = -2$ is a vertical asymptote.
Step 3:
To determine horizontal asymptotes, consider the degrees of the numerator and denominator in the function $R(x) = \frac{ax^n}{bx^m}$. The rules are as follows:
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 4:
Calculate the degrees of the numerator and denominator. For the given function, $n = 2$ and $m = 2$.
Step 5:
Since the degrees are equal ($n = m$), the horizontal asymptote is found using $y = \frac{a}{b}$. Here, $a = -1$ and $b = 5$, so the horizontal asymptote is $y = -\frac{1}{5}$.
Step 6:
An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of asymptotes:
- Vertical Asymptotes: $x = -2$
- Horizontal Asymptotes: $y = -\frac{1}{5}$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the different types of asymptotes:
Vertical Asymptotes: These occur at values of $x$ where the function is undefined, typically where the denominator is zero. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on 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 $y = \frac{a}{b}$, 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. The oblique asymptote can be found by performing polynomial long division.
Undefined Points: These are specific values of $x$ for which the function does not have a real value. They are found by setting the denominator equal to zero and solving for $x$.
Behavior Around Asymptotes: To determine the behavior of the function around vertical asymptotes, one must analyze the limits of the function as $x$ approaches the asymptote from the left and right.
In LaTeX, to write fractions, use the $\frac{numerator}{denominator}$command. For powers, use the caret symbol $^$followed by the exponent in curly braces ${}$. For example, $x^2$ is written as $x^2$.
