Find the Asymptotes (x^2)/((x-4)^2)
The problem is asking for the identification of asymptotes in the context of the function f(x) = (x^2)/((x-4)^2). Asymptotes are lines that a graph approaches but never actually reaches. They can be vertical, horizontal, or oblique (slant). The function given is a rational function, which often has asymptotes where the denominator is zero (leading to vertical asymptotes) or where the degrees of the polynomial in the numerator and denominator dictate the behavior at infinity (horizontal or oblique asymptotes). The task involves finding these asymptotes by analyzing the function's behavior as x approaches certain critical values or infinity.
$\frac{x^{2}}{\left(\left(\right. x - 4 \left.\right)\right)^{2}}$
Answer
Solution:
Step 1:
Determine the points at which the function $\frac{x^{2}}{(x-4)^{2}}$ is not defined. This occurs when $x = 4$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$ where $n$ represents the degree of the numerator and $m$ represents the degree of the denominator. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but potentially an oblique asymptote.
Step 3:
Identify the values of $n$ and $m$. For our function, $n = 2$ and $m = 2$.
Step 4:
Since $n$ equals $m$, the horizontal asymptote can be found using $y = \frac{a}{b}$. Given that $a = 1$ and $b = 1$, the horizontal asymptote is $y = 1$.
Step 5:
An oblique asymptote does not exist for this function because the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = 4$
- Horizontal Asymptotes: $y = 1$
- There are no Oblique Asymptotes.
Step 7:
Knowledge Notes:
Asymptotes are lines that a graph approaches but never actually touches or crosses. They can be vertical, horizontal, or oblique (slant).
Vertical Asymptotes occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero.
Horizontal Asymptotes are determined by comparing the degrees of the numerator ($n$) and denominator ($m$) in a rational function:
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 may occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, you would perform polynomial long division to find the equation of the oblique asymptote.
Rational Functions are functions expressed as the ratio of two polynomials. The degrees of the polynomials in the numerator and denominator can provide information about the behavior of the graph at infinity and the existence of asymptotes.
Undefined Points in the context of rational functions are the values of $x$ for which the function does not produce a real number output, typically where the denominator is zero.
Leading Coefficients are the coefficients of the highest degree terms in a polynomial. They play a significant role in determining the horizontal asymptote of a rational function when the degrees of the numerator and denominator are equal.
