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.
y=4x^2-100/2x^2+x-15
This first thing we need to do is 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 2x-5=0.
ccc
x+3=0 & 2x-5=0
⇕ & ⇕
x=-3 & x= 52
This means that x=- 3 and x= 52 are not included in the domain.
Once again, let's consider the given function.
y=4x^2-100/2x^2+x-15
Notice that the numerator can be factored. Let's factor this expression and try to cancel out any common factors between the numerator and the denominator.
Recall that we already factored the denominator. Let's look at fully factored fraction.
y=4(x+5)(x-5)/(x+3)(2x-5)
Note that we cannot cancel out common factors. Therefore, there are no holes. Also, if the real number a is not included in the domain, there is a vertical asymptote at x=a. In this case, we have a vertical asymptote at x=- 3 and x= 52.
Horizontal Asymptotes
To find the horizontal asymptotes, we can use a set of rules. To properly use these rules, in the following table m and n must be the highest degree of the numerator and denominator.
y=ax^m/bx^n
Asymptote
m
y=0
m>n
none
m=n
y=a/b
Now, consider the function one more time. Let's look at the degrees of the numerator and denominator for our function.
y=4x^2-100/2x^2+x-15
We can see that the degree of the denominator is equal to the degree of the numerator. Therefore, the line y= 42=2 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 intersects the x-axis. At these points, the value of the y-coordinate is zero. Let's substitute 0 for y in the given function and solve for x.
The y-intercept is the point where the graph intersects 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 solve for y.
