00:01
This problem gives us the function f of x equals 2x squared minus 11x plus 15 over 2x squared minus 7x minus 4.
00:07
And we want to find the y -intercept for this function, the x -intercepts, and the vertical asymptotes for this rational function.
00:13
And first for the y -intercept, that is a point where we always have zero for the x value, and then whatever y value we're crossing the y -axis at.
00:20
And we could evaluate our function at zero, but there's a trick about rational functions where you can just look at the constants at the end of your function and just take the ratio of those, because when we plug in zero for all of our x values, zero squared will be zero, and zero times anything will be zero, so we're just going to be left with 15 over negative 4.
00:42
So that would be zero, negative 15, fourths as our y -intercept.
00:47
Now to find our x -intercepts as well as our vertical asymptotes, we're going to do something very similar, but we're going to do it with the numerator and denominator depending on which one we're trying to find.
00:56
If we're trying to find x -intercepts, that occurs when our numerator turns to zero, because zero divided by a value is still zero.
01:03
So we can solve for these x -intercepts by factoring, and we can't factor out our 2, so we have to multiply this a value times c, which gives us positive 30.
01:11
And then we ask ourselves what two numbers multiply to be positive 30, but add to be negative 11.
01:15
And those two numbers are negative 6 and negative 5.
01:19
But since we multiplied by a to start our factoring, we have to divide by a now, and that gives us the factored form of x minus 3, and then 2x minus 5, after 5 halves won't simplify and we have to take the 2 in front.
01:33
So when this factored form is set equal to zero, we get two solutions...