00:01
This problem we're asked to graph the rational function 3x squared minus 5x minus 2 divided by x squared plus 1.
00:05
And we want to find all the asymptotes that exist for this function as well as some additional characteristics and points to help us make our graph.
00:12
So we'll go through the list of our asymptotes first.
00:15
Vertical asymptotes would come from solutions of the denominator set equal to 0 and solved.
00:19
But if we set this equal to 0 and solve and subtract the 1 over, we're going to end up taking the square root of a negative number and that's imaginary.
00:26
So we have no vertical asymptotes here.
00:28
For slant asymptotes, that's a pretty specific case where the degree of the denominator has to be exactly one less than a numerator so you can do division.
00:37
But that's not the case here, so there's no slant.
00:39
There is a horizontal asymptote because our degrees match up and when that happens, you take the ratio of the coefficients, so 3 over understood 1.
00:48
And that gives us y equals 1 for our horizontal asthmat.
00:51
Excuse me, that's y equals 3.
00:53
So we'll be able to draw in our horizontal asymptotes as best we can.
01:02
And then we can continue with the graph.
01:04
So we can now find additional points, but we can also find more additional points before just doing a t -chart to just find additional ones by looking at the function.
01:13
You can find x intercepts by setting the numerator equal to zero.
01:17
And this could factor, you could solve it in the calculator.
01:20
But no matter what, your zeros will come out to be x equals negative 1.
01:26
1 3rd, and 2, so we can plot those on our graph as best we can...