Recall that division by zero is not defined. This means 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=3x/x-4
Recall that division by zero is not defined. Therefore, the rational function is undefined where x-4=0
x-4=0 ⇔ x=4
The above means that x=4 is not included in the domain.
Consider the given function.
y=3x/x-4
Note that we cannot cancel out common factors. Therefore, there are no holes and if a 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=4.
Horizontal Asymptote
Let's pay close attention to the degrees of the numerator and denominator.
y=3x^1/x^1-4
We see that the degrees of the numerator and denominator are the same. To find the horizontal asymptote, we need to find the quotient between the leading coefficients.
y=3x^1/1x^1-4
Since 31=3, there is a horizontal asymptote at y=3.
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.
