Find the Asymptotes (x^2+5x+6)/(x+3)
The question asks for the identification of asymptotes of the given rational function. An asymptote is a line that the graph of a function approaches as the value of a variable (usually x) becomes very large or very small (tends towards positive or negative infinity). There are different types of asymptotes, such as vertical, horizontal, and oblique (or slant) asymptotes.
For the function (x^2+5x+6)/(x+3), you're expected to find any lines that the graph of this function approaches but never crosses as x approaches infinity or negative infinity, or places where the function is undefined (potential vertical asymptotes).
To do this, you need to look at the behavior of the function as x approaches certain critical values and evaluate the limits at infinity or negative infinity. For vertical asymptotes, you'll be interested in the points where the denominator equals zero and for horizontal or slant asymptotes, you'll be considering the end behavior of the function as x approaches infinity or negative infinity.
$\frac{x^{2} + 5 x + 6}{x + 3}$
Answer
Solution:
Step:1
Determine the points where the function $\frac{x^{2} + 5x + 6}{x + 3}$ is not defined, which are the values that make the denominator zero. In this case, $x = -3$.
Step:2
Identify the vertical asymptotes, which are typically where the function is undefined due to division by zero. However, in this case, there are No Vertical Asymptotes.
Step:3
To find horizontal asymptotes, consider a general 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 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, but there may be an oblique asymptote.
Step:4
Calculate the degrees $n$ and $m$ for the given function. Here, $n = 2$ and $m = 1$.
Step:5
Since the degree of the numerator is greater than the degree of the denominator ($n > m$), there is No Horizontal Asymptotes.
Step:6
To find the oblique asymptote, perform polynomial long division.
Step:6.1
Simplify the given expression.
Step:6.1.1
Factor the numerator $x^{2} + 5x + 6$ using the AC method.
Step:6.1.1.1
Look for two integers whose product equals the constant term $c$ and whose sum equals the coefficient $b$. In this case, find integers with a product of $6$ and a sum of $5$. The integers are $2$ and $3$.
Step:6.1.1.2
Express the factored form with these integers as $\frac{(x + 2)(x + 3)}{x + 3}$.
Step:6.1.2
Eliminate the common factor in the numerator and denominator.
Step:6.1.2.1
Remove the common factor to get $\frac{(x + 2)\cancel{(x + 3)}}{\cancel{(x + 3)}}$.
Step:6.1.2.2
Divide the remaining term $x + 2$ by $1$ to obtain $x + 2$.
Step:6.2
The oblique asymptote is given by the quotient of the division, which is $y = x + 2$.
Step:7
Compile the complete set of asymptotes for the function:
- No Vertical Asymptotes
- No Horizontal Asymptotes
- Oblique Asymptote: $y = x + 2$
Step:8
Knowledge Notes:
Asymptotes are lines that the graph of a function approaches but never touches. They can be vertical, horizontal, or oblique (slant).
Vertical Asymptotes occur at values of $x$ that make the denominator of a rational function zero, provided the numerator does not also become zero at those points.
Horizontal Asymptotes are found by comparing the degrees of the numerator and denominator. If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the x-axis is the horizontal asymptote. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients. If $n > m$, there is no horizontal asymptote.
Oblique Asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They can be found by performing polynomial long division and taking the quotient as the equation of the asymptote.
Polynomial Long Division is a method used to divide polynomials, similar to the long division of numbers. It can be used to simplify rational functions and find oblique asymptotes.
Factoring is a process of breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. The AC method is one such factoring technique used for quadratic expressions.
Cancelling Common Factors in the numerator and denominator of a rational expression can simplify the expression and reveal more information about the function, such as its asymptotes or simplified form.
