Find the Asymptotes f(x)=(4x^2-25x+6)/(x^2-5x-6)
The given problem asks to determine the vertical and horizontal (or possibly oblique) asymptotes of the function f(x) = (4x^2 - 25x + 6) / (x^2 - 5x - 6). Asymptotes are lines that the graph of the function approaches but never actually reaches as x goes to infinity or to certain critical values.
Vertical asymptotes typically occur at values of x for which the denominator of the function equals zero, provided these aren't canceled out by the same factors in the numerator.
Horizontal or oblique asymptotes are found by examining the end behavior of the function as x approaches infinity or negative infinity. They give information about how the function behaves at the extreme ends of the x-axis. The existence and nature of these asymptotes depend on the degrees of the polynomial in the numerator and the denominator.
$f \left(\right. x \left.\right) = \frac{4 x^{2} - 25 x + 6}{x^{2} - 5 x - 6}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which $\frac{4x^2 - 25x + 6}{x^2 - 5x - 6}$ does not exist. These are $x = -1$ and $x = 6$.
Step 2:
As $x$ approaches $-1$ from the left, $\frac{4x^2 - 25x + 6}{x^2 - 5x - 6}$ tends towards positive infinity, and from the right, it tends towards negative infinity. Therefore, $x = -1$ is a vertical asymptote.
Step 3:
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, the horizontal asymptote depends on the relationship between $n$ and $m$. If $n < m$, the horizontal asymptote is $y = 0$. If $n = m$, it is $y = \frac{a}{b}$. If $n > m$, no horizontal asymptote exists (instead, there may be an oblique asymptote).
Step 4:
Identify the degrees $n$ and $m$ of the numerator and denominator, respectively. Here, $n = 2$ and $m = 2$.
Step 5:
Given that $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, with $a = 4$ and $b = 1$. Thus, the horizontal asymptote is $y = 4$.
Step 6:
There is no oblique asymptote since the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of asymptotes:
- Vertical Asymptotes: $x = -1$
- Horizontal Asymptotes: $y = 4$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the different types of asymptotes and how they relate to the function's structure:
Vertical Asymptotes: These occur at values of $x$ where the denominator of the rational function is zero (assuming the numerator is not also zero at these points). They represent values where the function tends towards infinity.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator ($n$) and the 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 greater than the degree of the denominator. The oblique asymptote can be found by performing polynomial long division.
Rational Functions: A rational function is a ratio of two polynomials, written in the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Behavior at Infinity: To determine the behavior of a rational function as $x$ approaches infinity or negative infinity, one can use the leading terms of the numerator and denominator.
Polynomial Division: When the degree of the numerator is greater than the degree of the denominator, polynomial long division can be used to find the oblique asymptote.
Understanding these concepts is crucial for analyzing the behavior of rational functions and identifying their asymptotes.
