To graph the given , we will find its , , and . Then, we will find points using a . Finally, we will plot and connect those points.
Domain
Consider the given function.
y=x^2+3x-6/x-2
Recall that division by zero is not defined. Therefore, the rational function is undefined where x-2=0.
x-2=0 ⇔ x=2
This means that x=2 is not included in the domain.
Domain
All real numbers except x=2
Asymptotes
Asymptotes can be , , or oblique lines.
Vertical Asymptotes
Once again, let's consider the given function.
y=x^2+3x-6/x-2
In this case, the numerator cannot be factored and we cannot cancel out common factors. This means that there are no holes. Therefore, 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=2.
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 are the highest of the numerator and denominator.
| y=ax^m/bx^n
|
Asymptote
|
Asymptote Type
|
| m
| y=0
|
Horizontal
|
| m>n and m-n≠1
|
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^2+3x-6/x^1-2
Here, the degree of the numerator minus the degree of the denominator equals 1. Therefore, the function has an oblique asymptote. The equation of this asymptote is found by dividing the numerator by the denominator, with no remainder.
l r x -2 & |l x^2+ 3x - 6
r x r x-2 & |l x^2+3x-6
r x rl x-2 & |l x^2+3x-6 & x^2 -2x
r x r x-2 & |l 5x -6
r x+5 r x-2 & |l 5x-6
rx+ 5 rl x-2 & |l 5x-6 & 5x-10
r x+5 r x-2 & |l 4
The quotient is x+5, with remainder 4.
The equation of the oblique asymptote is y=x+5.
Intercepts
The of the function are the points at which the graph intersects the axes.
x-intercepts
The 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.
y=x^2+3x-6/x-2
0=x^2+3x-6/x-2
x^2+3x-6=0
Note that we have a now. Let's identify the values of a, b, and c.
x^2+3x-6=0
⇕
1x^2 +3x+(- 6)=0
We have that a= 1, b=3, and c=-6. Let's substitute these values into the and solve for x.
x=- b±sqrt(b^2-4ac)/2a
x=- 3±sqrt(3^2-4( 1)(-6))/2( 1)
x=- 3±sqrt(33)/2
Let's calculate both solutions by using the positive and negative signs.
| x=- 3± sqrt(33)/2
|
| x=- 3+ sqrt(33)/2
|
x=- 3- sqrt(33)/2
|
| x≈ - 3+5.74/2
|
x≈ - 3-5.74/2
|
| x≈ 1.37
|
x≈ - 4.37
|
There are x-intercepts around (1.37,0) and (- 4.37,0).
y-intercept
The 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.
y=x^2+3x-6/x-2
y=0^2+3( 0)-6/0-2
y=3
There is a y-intercept at (0,3).
Graph
Let's make a table of values to graph the given function. Make sure to only use values included in the domain of the function.
| x
|
x^2+3x-6/x-2
|
y=x^2+3x-6/x-2
|
| 3
|
3^2+3( 3)-6/3-2
|
12
|
| 4
|
4^2+3( 4)-6/4-2
|
11
|
| 6
|
6^2+3( 6)-6/6-2
|
12
|
Finally, let's plot and connect the points. Do not forget to draw the asymptote and to plot the intercepts.
