00:01
So we've been given a function f of x where we have on the numerator a quadratic and another quadratic on the denominator.
00:09
In order to draw this graph, we need to simplify things, make this as neat and easy as possible to deal with.
00:16
So to do that, we're going to have to spot some factors on the top and the bottom.
00:20
Instantly, you can realize the sort of spot on the bottom.
00:24
We've got a difference of two squares.
00:26
So let's write that down.
00:27
So this is going to be x minus 3 and x plus 3 on the bottom.
00:32
And then on the top, playing around and sort of seeing factors of 12 and making sure we account for the minus, we see that x minus 4 and x minus 3, oh sorry, x minus 4 and x plus 3 work.
00:48
So here we have a cancellation of both the x plus 3 and the numerator and denominator.
00:56
So we can simplify this as follows into x minus 4 over x minus 3.
01:07
We can simplify this even further just to make this as easy as possible to deal with using long division.
01:13
So if we take x minus 3 as our, sorry, x minus 4 as our dividend, and then underneath sort of the, if you want to call it the bus, you have x minus 3.
01:31
So how many x -4 is going to x -3? only 1, so we get x -4 here.
01:42
We take them away from both sides, our x's cancel and then we end up with just 1 as our quotient or our remainder.
01:50
So putting this all together we get x -4 over x -3 equals 1 minus 1 over x -3.
02:03
Now being in this form will help us out heavily just a little bit later when we sketch the graph so first of all we need to find the x -intercept so i'll just write that out in red and straight after we'll need to find the y -intercept now these occur the x -intercept occurs when f of x is equal to 0 and the y -intercept occurs when x is equal to 0 so let's sub in these two accordingly when f of x is equal to 0 we get 1 minus 1 over x minus 3 equals 0 and then doing some rearranging you should get 1 equals 1 over x minus 3 which gives us just x minus 3 on the other side so x minus 3 and so our intercept is x equals 4 so so just removing all this and going back, you can just write x equals 4.
03:24
So our coordinates will be 4, 0.
03:29
Now when x equals 0 in f of x, we sub in 0.
03:34
So we get f of 0 equals 1 minus 1 over 0 minus 3, which is just minus 3.
03:41
So let's just write that.
03:45
The minuses here are going to make a plus.
03:47
So we get 1 plus 1 over 3, which is just 4 over 3.
03:53
So our coordinates for the x -intercept is 4, 0, and the coordinates for the y -intercept are going to be 0, 4 over 3.
04:05
Okay, so now we have our y and x -intercept...