Understanding and Drawing Cubic Graphs

In this video we're going to look at how to draw graphs of cubics. So here we've got the general equation of a cubic y equals ax cubed plus bx squared plus Cx plus D where the a cannot equal zero in other words it's going to have to have an x cubed term its highest x term. Here's some examples of cubics. y equals x cubed. y equals minus x cubed. y equals 2x cubed plus ax squared minus 7x plus 3. So as you can see you can have an x squared term an x and a 3. or number on the end, but obviously it's going to have to have that x cubed term. y equals x cubed plus 4x minus 2, so it doesn't have to have that x squared term. So it's just any equation, y equals, that's got an x cubed, and it may or may not have stuff after it, so that's the cubic graph. Okay, so let's start off by looking at y equals x cubed. So this is the y equals x cubed graph. It comes up like so, it flattens off at zero, and then comes up again in a curve. fashion like so very steeply um let's have a look and see why so let's have a look at uh to the right of the y-axis so let's start off with zero well zero cubed zero times zero times zero is zero so that's why you'll have the graph going through the origin okay so y equals x cubed will pass through the origin next let's try one well one cubed one times one times one is one so that'd be one across one up let's try two two times two times two is eight so we're going two across eight up so it's going quite high if we try three well three times times 3 times 3 is 27 so that's 3 across 27 up and if we have a look at 4 4 times 4 times 4 well that's 64 so as you can see it goes up very quickly okay it's a very steep curve going up like so next let's have a look at the left of the y-axis okay so uh we've tried zero and that's the origin okay let's try minus 1 well minus 1 times minus 1 times minus 1 well negative times a negative is a positive times a negative is a negative so minus 1 times minus 1 times minus 1 would be minus 1 Minus 2 times minus 2 times minus 2. Well, minus 2 times minus 2 is 4. Times minus 2 is minus 8. And then minus 3, you'll see, this will be familiar, will be minus 27. Minus 3 times minus 3 is 9. Times minus 3 is minus 27. And so on. So it's similar, but it's going downwards, okay? It's below the x-axis. So 1 to the left, 1 down. 2 to the left, 8 down, and so on. So again, it goes down very steeply. And that's it. So that's the y equals x cubed graph. It comes up flat. off and passes through the origin and curves up again. Okay so this is the y equals negative x cubed graph so comes down curves down like so, flattens off passes through the origin and curves downwards again very sharply. Let's have a look and see why. So 0 cubed 0 times 0 times 0 is 0 but negative so still 0. 1 cubed is 1 but negative so minus 1. 2 cubed 8 again negative minus 8 so again it's coming across 1 across 1 down 2 across 8 down and so on. so it's going to be negative 27. Negative, so negative 1 cubed is negative 1, but negative, so minus 1 times bma positive would be 1, or times by minus 1. Negative 2 cubed, well, it's going to be minus 8. Again, that's then going to become 8, and 27. So as you can see, that's why it's coming down like so, and passing through the origin and down again. Okay, so that's the two very common graphs, y equals x cubed. And... and y equals negative x cubed. Now, not all the x cubed graphs will have that shape. Let's have a look at some other ones, okay? Those ones are very simple x cubed graphs. The reason is they don't have any x squared x's or constant terms. Okay, so let's have a look at how to draw the graph y equals x cubed plus 2x squared minus 3. So you can see we've got some extra terms here. Okay, we don't just have the x cubed term. We've got an x squared term and a constant on the end. So I've made a table of values, and the table of values starts at minus 3, and it goes... to 2. So first of all I usually start these by looking at the positive values so let's start off with 2. Okay so 2 cubed well 2 cubed is going to be 8 and then plus and then 2 squared well 2 well I'm using bot maths here I'm going to square the numbers and then times them by 2. Okay so 2 squared is 4 times by 2 is 8 and then I'm going to take away 3 and when I do that I get an answer of 13. Okay because 8 plus 8 is equal to 16 minus 3 is equal to 13. So that's going to be that coordinate. So that's 2 across 13 up. So that's that point there. Next, 1. So again, 1 cubed. Well, 1 cubed is equal to 1. 1 squared is equal to 1, but times by 2 will be plus 2, and then take away 3. Well, that's going to be 1 plus 2 is equal to 3. Take away 3 is equal to 0. So that point there is equal to 0. So that's 1 across, 0 up. Okay? Next, 0. Okay, so 0 cubed. 0 plus 0 squared of 0 times 2 is plus 0 and take away 3 that's going to give us negative 3. So that's going to be 0 across 3 down. Next minus 1. Now we're going to be very careful with these negatives. okay so minus 1 well minus 1 times minus 1 times minus 1 well that's going to be minus 1 then we're going to square it so minus 1 squared is 1 and times by 2 is 2 so plus 2 and then take away 3 minus 1 plus 2 is going to be equal to 1 take away 3 is going to be equal to negative 2 so it's negative 2 so that's 1 across minus 2 down so there next minus 2 so again be very careful minus 2 cubed is going to be minus 8 again we're going to square it first so minus 2 times minus 2 is equal to 4 times 2 is equal to 8 so it's plus 8 and then take away 3 and that's going to give us an answer of minus 3 so it's going to be the point minus 2 minus 3 so it's minus 2 minus 3 it's going to move the point slightly Okay, and our last one, minus 3. So minus 3 cubed is going to be equal to minus 27. Plus, well remember to square it, so minus 3 squared is 9, times 2 is plus 18. So it's going to be minus 9. Take away 3, it's going to be equal to minus 12. That's it. So let's plot that point to minus 3 minus 12 As you can see the points here come up They then sort of curve and flatten out here and then reach the house point there And then curve back down again and then come back up again. Okay So it sort of comes up and then down and then up again now I'm going to try and draw this now this won't be the best because I'm trying to draw on a tablet, okay? So comes up Flattens comes down again up through here comes up and then then curves upwards like so. Now I have prepared one of these earlier so I'm going to show you what it looks like properly. It comes up and then down and then up again. So it's got a maximum and a minimum. It comes up, flattens at this maximum point, down again at this local minimum point and then it curves up again. Let's have a look at what it looks like. So this is what it looks like. Okay, so here's a slightly better version. As you can see, it comes up, flattens off, comes down again, curves, flattens off again, and then up again like so. So that's quite a common cubic graph shape where it comes up, and then down, and then up again. Okay, let's have a look at it. our last example so our last example is to draw y equals minus x cubed plus 3x squared plus x minus 1 and what we're going to do here is and the main difference here is i've put a negative sign in front of the x cubed okay i'm going to start with the positives again okay so let's start off with 4 so we're going to do uh well uh 4 cubed well 4 cubed is 64 but it's minus so it's going to be minus 64 and then remembering bob mass what we're going to do is we're going to square the 4 which is equal to 16 but then we're going to times it by 3 so it'll be plus 48 and then we're going to plus 4 and then minus 1 and when we do that we get an answer off if you work that out you get an answer of minus 13 so that's going to be 4 minus 13 next one 3 so minus 3 cubed well 3 cubed 27 so it's going to be minus 27 again remember in Bobmas 3 squared is 9 and then times by 3 is then going to be plus 27 and then plus X so that would be plus 3 and then minus 1 and that gives us an answer of 2 so that's 2. Let's now work out 2 well whenever you work out 2 using the same approach you're going to get 2 cubed so that's going to be minus 8 and then plus again Bobmas 2 squared is going to be 4 times 3 is going to be plus 12 and then we're going to add on 2 and we're going to going to then take away 1 when we do that we're going to get 4 6 5 so it's going to be equal to 5 1 well it's going to be equal to minus 1 plus 3 plus 1 and then minus 1 that gives us an answer of 2 so it's going to be 2 plus 1 minus 1 it's going to be an answer of 2 there 0 putting in 0 is going to give us 0 plus 0 plus 0 plus minus 1 so it'll be minus 1 now again let's be very careful with these negatives okay so minus 1 cubed okay so minus 1 cubed is going to be equal to minus 1 but minus is going to be 1 and then plus and then minus 1 squared is 1 and then times 3 is going to be plus 3 and then plus minus 1 and then minus 1 whenever you work that out it's going to be 4 minus 1 which is 3 minus 1 which is 2 next one minus 2 so it's going to be minus 2 cubed which is minus 8 but minus it's going to be 8 and then plus minus 2 squared is going to be minus 2 squared is going to be 4 and then times 3 is going to be plus 12 and then plus minus 2 and then minus 1 whenever you work that out what we're going to get is an answer of 17 that's going to be 20 minus 2 is 18 minus 1 is equal to 17 so it's going to be 17 and then minus 3, whenever you work that out you're gonna get that's equal to 50. Okay so let's plot our points. So 4 minus 13, well we can't plot that on this graph, but we're gonna know it's gonna be minus 4, sorry 4 minus 14 is gonna be down here somewhere okay. next one three two so three twos up here so it's going to come up here and then it's going to reach two five so two and then five next is equal the point one two so one two and then we've got zero minus one so zero minus ones there and the next one minus one two and the next one is minus 217 so it's going to be up here somewhere and then minus 350 so if you were to draw this now obviously sort of using a bit of imagination it's going to come down like so curving down and then it's going to come flat in there and then up again and then it's going to curve and it's going to come down again and then down quite sharply okay so it's quite similar to our previous one except for rather than coming up and then down and then up again this one's coming down and then up and then down again a slightly better version is this version here Okay, so as you can see it comes down Which is the turning point so it sort of then comes back up again, which is another turning point and comes back down again Okay, and that's it. So that's how you draw cubic graphs. These are the common shapes So you've got your x cubed one You've got your minus x cubed one and you've got these versions which are very common This one will be for a positive x cubed and then it would have some terms after it perhaps I would come up and then down and up again and this one will be for minus x cubed and then I would have some terms after it and and it would come down and then up and then down again. And that's it, so that's qubit graphs.

y equals x cubed. y equals minus x cubed. y equals 2x cubed plus ax squared minus 7x plus 3. So as you can see you can have an x squared term an x and a 3. or number on the end, but obviously it's going to have to have that x cubed term.

y equals x cubed plus 4x minus 2, so it doesn't have to have that x squared term. So it's just any equation, y equals, that's got an x cubed, and it may or may not have stuff after it, so that's the cubic graph. Okay, so let's start off by looking at y equals x cubed.

So this is the y equals x cubed graph. It comes up like so, it flattens off at zero, and then comes up again in a curve. fashion like so very steeply um let's have a look and see why so let's have a look at uh to the right of the y-axis so let's start off with zero well zero cubed zero times zero times zero is zero so that's why you'll have the graph going through the origin okay so y equals x cubed will pass through the origin next let's try one well one cubed one times one times one is one so that'd be one across one up let's try two two times two times two is eight so we're going two across eight up so it's going quite high if we try three well three times times 3 times 3 is 27 so that's 3 across 27 up and if we have a look at 4 4 times 4 times 4 well that's 64 so as you can see it goes up very quickly okay it's a very steep curve going up like so next let's have a look at the left of the y-axis okay so uh we've tried zero and that's the origin okay let's try minus 1 well minus 1 times minus 1 times minus 1 well negative times a negative is a positive times a negative is a negative so minus 1 times minus 1 times minus 1 would be minus 1 Minus 2 times minus 2 times minus 2. Well, minus 2 times minus 2 is 4. Times minus 2 is minus 8. And then minus 3, you'll see, this will be familiar, will be minus 27. Minus 3 times minus 3 is 9. Times minus 3 is minus 27. And so on. So it's similar, but it's going downwards, okay? It's below the x-axis.

So 1 to the left, 1 down. 2 to the left, 8 down, and so on. So again, it goes down very steeply.

And that's it. So that's the y equals x cubed graph. It comes up flat. off and passes through the origin and curves up again.

Okay so this is the y equals negative x cubed graph so comes down curves down like so, flattens off passes through the origin and curves downwards again very sharply. Let's have a look and see why. So 0 cubed 0 times 0 times 0 is 0 but negative so still 0. 1 cubed is 1 but negative so minus 1. 2 cubed 8 again negative minus 8 so again it's coming across 1 across 1 down 2 across 8 down and so on. so it's going to be negative 27. Negative, so negative 1 cubed is negative 1, but negative, so minus 1 times bma positive would be 1, or times by minus 1. Negative 2 cubed, well, it's going to be minus 8. Again, that's then going to become 8, and 27. So as you can see, that's why it's coming down like so, and passing through the origin and down again.

Okay, so that's the two very common graphs, y equals x cubed. And... and y equals negative x cubed.

Now, not all the x cubed graphs will have that shape. Let's have a look at some other ones, okay? Those ones are very simple x cubed graphs.

The reason is they don't have any x squared x's or constant terms. Okay, so let's have a look at how to draw the graph y equals x cubed plus 2x squared minus 3. So you can see we've got some extra terms here. Okay, we don't just have the x cubed term. We've got an x squared term and a constant on the end. So I've made a table of values, and the table of values starts at minus 3, and it goes...

to 2. So first of all I usually start these by looking at the positive values so let's start off with 2. Okay so 2 cubed well 2 cubed is going to be 8 and then plus and then 2 squared well 2 well I'm using bot maths here I'm going to square the numbers and then times them by 2. Okay so 2 squared is 4 times by 2 is 8 and then I'm going to take away 3 and when I do that I get an answer of 13. Okay because 8 plus 8 is equal to 16 minus 3 is equal to 13. So that's going to be that coordinate. So that's 2 across 13 up. So that's that point there.

Next, 1. So again, 1 cubed. Well, 1 cubed is equal to 1. 1 squared is equal to 1, but times by 2 will be plus 2, and then take away 3. Well, that's going to be 1 plus 2 is equal to 3. Take away 3 is equal to 0. So that point there is equal to 0. So that's 1 across, 0 up. Okay? Next, 0. Okay, so 0 cubed. 0 plus 0 squared of 0 times 2 is plus 0 and take away 3 that's going to give us negative 3. So that's going to be 0 across 3 down.

Next minus 1. Now we're going to be very careful with these negatives. okay so minus 1 well minus 1 times minus 1 times minus 1 well that's going to be minus 1 then we're going to square it so minus 1 squared is 1 and times by 2 is 2 so plus 2 and then take away 3 minus 1 plus 2 is going to be equal to 1 take away 3 is going to be equal to negative 2 so it's negative 2 so that's 1 across minus 2 down so there next minus 2 so again be very careful minus 2 cubed is going to be minus 8 again we're going to square it first so minus 2 times minus 2 is equal to 4 times 2 is equal to 8 so it's plus 8 and then take away 3 and that's going to give us an answer of minus 3 so it's going to be the point minus 2 minus 3 so it's minus 2 minus 3 it's going to move the point slightly Okay, and our last one, minus 3. So minus 3 cubed is going to be equal to minus 27. Plus, well remember to square it, so minus 3 squared is 9, times 2 is plus 18. So it's going to be minus 9. Take away 3, it's going to be equal to minus 12. That's it. So let's plot that point to minus 3 minus 12 As you can see the points here come up They then sort of curve and flatten out here and then reach the house point there And then curve back down again and then come back up again.

Okay So it sort of comes up and then down and then up again now I'm going to try and draw this now this won't be the best because I'm trying to draw on a tablet, okay? So comes up Flattens comes down again up through here comes up and then then curves upwards like so. Now I have prepared one of these earlier so I'm going to show you what it looks like properly.

It comes up and then down and then up again. So it's got a maximum and a minimum. It comes up, flattens at this maximum point, down again at this local minimum point and then it curves up again. Let's have a look at what it looks like. So this is what it looks like.

Okay, so here's a slightly better version. As you can see, it comes up, flattens off, comes down again, curves, flattens off again, and then up again like so. So that's quite a common cubic graph shape where it comes up, and then down, and then up again.

Okay, let's have a look at it. our last example so our last example is to draw y equals minus x cubed plus 3x squared plus x minus 1 and what we're going to do here is and the main difference here is i've put a negative sign in front of the x cubed okay i'm going to start with the positives again okay so let's start off with 4 so we're going to do uh well uh 4 cubed well 4 cubed is 64 but it's minus so it's going to be minus 64 and then remembering bob mass what we're going to do is we're going to square the 4 which is equal to 16 but then we're going to times it by 3 so it'll be plus 48 and then we're going to plus 4 and then minus 1 and when we do that we get an answer off if you work that out you get an answer of minus 13 so that's going to be 4 minus 13 next one 3 so minus 3 cubed well 3 cubed 27 so it's going to be minus 27 again remember in Bobmas 3 squared is 9 and then times by 3 is then going to be plus 27 and then plus X so that would be plus 3 and then minus 1 and that gives us an answer of 2 so that's 2. Let's now work out 2 well whenever you work out 2 using the same approach you're going to get 2 cubed so that's going to be minus 8 and then plus again Bobmas 2 squared is going to be 4 times 3 is going to be plus 12 and then we're going to add on 2 and we're going to going to then take away 1 when we do that we're going to get 4 6 5 so it's going to be equal to 5 1 well it's going to be equal to minus 1 plus 3 plus 1 and then minus 1 that gives us an answer of 2 so it's going to be 2 plus 1 minus 1 it's going to be an answer of 2 there 0 putting in 0 is going to give us 0 plus 0 plus 0 plus minus 1 so it'll be minus 1 now again let's be very careful with these negatives okay so minus 1 cubed okay so minus 1 cubed is going to be equal to minus 1 but minus is going to be 1 and then plus and then minus 1 squared is 1 and then times 3 is going to be plus 3 and then plus minus 1 and then minus 1 whenever you work that out it's going to be 4 minus 1 which is 3 minus 1 which is 2 next one minus 2 so it's going to be minus 2 cubed which is minus 8 but minus it's going to be 8 and then plus minus 2 squared is going to be minus 2 squared is going to be 4 and then times 3 is going to be plus 12 and then plus minus 2 and then minus 1 whenever you work that out what we're going to get is an answer of 17 that's going to be 20 minus 2 is 18 minus 1 is equal to 17 so it's going to be 17 and then minus 3, whenever you work that out you're gonna get that's equal to 50. Okay so let's plot our points. So 4 minus 13, well we can't plot that on this graph, but we're gonna know it's gonna be minus 4, sorry 4 minus 14 is gonna be down here somewhere okay. next one three two so three twos up here so it's going to come up here and then it's going to reach two five so two and then five next is equal the point one two so one two and then we've got zero minus one so zero minus ones there and the next one minus one two and the next one is minus 217 so it's going to be up here somewhere and then minus 350 so if you were to draw this now obviously sort of using a bit of imagination it's going to come down like so curving down and then it's going to come flat in there and then up again and then it's going to curve and it's going to come down again and then down quite sharply okay so it's quite similar to our previous one except for rather than coming up and then down and then up again this one's coming down and then up and then down again a slightly better version is this version here Okay, so as you can see it comes down Which is the turning point so it sort of then comes back up again, which is another turning point and comes back down again Okay, and that's it.

So that's how you draw cubic graphs. These are the common shapes So you've got your x cubed one You've got your minus x cubed one and you've got these versions which are very common This one will be for a positive x cubed and then it would have some terms after it perhaps I would come up and then down and up again and this one will be for minus x cubed and then I would have some terms after it and and it would come down and then up and then down again. And that's it, so that's qubit graphs.

Transcript for:Understanding and Drawing Cubic Graphs

Transcript for:
Understanding and Drawing Cubic Graphs