in this lesson we are going to look at how we work out the equation of a cubic graph so all it is is you need to use the information that they give you and then plug it into the correct equation so for example if we are given the following then all you do is you take the information that they've given you and you plug it into the appropriate equation one by one so let's start with the first one they're telling us that they they want you to plug it into the graph of f which is this one they're saying that when x is zero the answer should be eight so we can do this okay so you see what i've done i've literally plugged in all the x as a zero and i've made the odds equal to eight because and that part cancels and that part cancels and so that allows us to find that d is 8. you see so we've already found one letter now now the next one says that we should plug the x value of 1 into the first derivative so let's go get the first derivative that's going to give us 3 a x squared plus 2 b x plus c and then the d would fall away they tell us that the answer is minus 5 when x is one so you see guys what i'm doing i'm literally just plugging in i'm not even thinking about this one i'm just plugging things in and then you simplify as far as possible and unfortunately there's no other on well we can't get any answers from that so we're going to plug an x value of 2 into the second derivative so the second derivative is going to look let's just write down the first derivative again so it's 3ax squared plus 2 b x plus c then the second derivative will be 6 a x plus 2 b they tell us that when x is 2 then the answer should be 8. that and then 8 is equal to 12 a plus 2 b we could divide everything by 2 there just to simplify it nicely there we go once again we can't get any more information but at least we've plugged it in so we've got now got two equations that we can use i'm just going to call them different things so that when i talk about them you know what i'm talking about we'll call that one a we'll call that one b and then we still have another piece of information to use so this one we're also going to plug into the second derivative on i've gone and taken it off again but what it was was 6 ax plus plus 2b and they're telling us that the answer should be 20 whenever we plug in an x value of 4 so we're going to plug in 4 and so that's going to give us 20 equals to 24a plus 2b i'm just going to divide everything by 2. you don't have to but it just makes life a bit easier and there we have another equation and so there we have it so have a look at this we're not going to look at this one over here just yet oh i don't want that to do that let's make that a triangle okay so we don't want it to we don't want to look at that one but if we look at these two it's two equations and there's two unknowns a and b so we just use a simultaneous equation over there so i'm going to take equation b and i'm just going to rearrange it to become 4 minus 6a then i'm going to take that expression and plug it in the place of b in the other equation so it's going to be 10 equals to 12 a plus then b is going to be replaced with 4 minus 6a there are other ways to do this many students like to get the b's by themselves that also works so then what we're going to get is 6 a on the right and 6 on the left so a is 1. i'm then going to plug that back over there so that b would be equal to 4 minus 6 times 1 so b is going to be equal to negative 2. now that we have a and b we can come back to this equation that was at the top and we just fill in the a value and we fill in the b value like that and then we just bring everything over to the left at -3 this is going to be -4 so on the left it will be plus 4 so that means c is going to be equal to -4 therefore we can say f of x is going to be equal to 1 x 3 you don't have to say 1 but yes you can if you want to minus 2x squared minus 4x and we said that d was 8 and there we have our equation so the most important thing is just go ahead and plug these into the appropriate equation and then solve