Division by zero is not defined. This means that the denominator cannot be zero.
Practice makes perfect
To graph the given rational function, we will find its domain, asymptotes, and intercepts. Then, we will find points using a table of values. Finally, we will plot and connect those points.
Domain
Consider the given function.
f(x)=x+4/x^2+7x+12
The first thing we need to do is to factor the denominator of this function.
Recall that division by zero is not defined. Therefore, the rational function is undefined where x+3=0 and x+4=0.
c|c
x+3=0 & x+4=0
⇕ & ⇕
x=-3 & x=-4
This means that x=- 3 and x=-4 are not included in the domain.
Domain
All real numbers except x=- 3 and x=-4
We canceled out the factor x+4. Therefore, there is a hole in the graph at x=-4. Since the factor x+3 is still in the denominator, we have a vertical asymptote at x=-3.
Horizontal and Oblique Asymptotes
To find the horizontal and oblique asymptotes, we can use a set of rules. To properly use these rules, in the following table, m and n must be the degree of the numerator and denominator.
y=ax^m/bx^n
Asymptote
Asymptote Type
m
y=0
Horizontal
m>n, with m-n≠1
none
none
m=n
y=a/b
Horizontal
m-n=1
y= the quotient of the polynomials with no remainder
Oblique
Now, consider the function one more time.
Let's look at the degrees of the numerator and denominator for our function.
y=x^1+4/x^2+7x+12
We can see that the degree of the denominator is higher than the degree of the numerator. Therefore, the line y=0 is a horizontal asymptote.
Intercepts
The intercepts of the function are the points at which the graph intersects the axes.
x-intercepts
The x-intercepts are the points where the graph intercepts the x-axis. At these points, the value of the y-coordinate is zero. Let's substitute 0 for f(x) in the given function and solve for x.
We found that if y=0, the value of x is -4. However, x=-4 is not included in the domain of the function. Therefore, there are no x-intercepts.
y-intercept
The y-intercept is the point where the graph intercepts the y-axis. At this point, the value of the x-coordinate is zero.
Let's substitute 0 for x in the given function and simplify.
