There are 4 commonly occurring graphs, these are; quadratic, cubic, reciprocal, exponential and circle graphs. Each of their equations can be used to plot a particular shape or curve on a graph.
A quadratic graph is produced when you have an equation of the form Y = ax^2 + bx + c, where b and c can be zero but a cannot be zero.
All quadratic graphs have a line of symmetry.
A cubic equation contains only terms up to and including x³.
A graph in the form of y = 1/x is known as a reciprocal graph.
Exponential graphs are found in the form of y = k^x. These graphs increase rapidly in the direction of y and will never fall below the x-axis.
We're looking at graphs today and there are four main types of graphs we're going to focus on. We've got quadratic. These are anything to do with X squared in them. We've got cubic, which is AI equation that has an X cubes in it. We've got reciprocal, which just means one over.
In this case, it's one over X. We could actually generalise this Kover X if we want to see. And finally we've got exponential. So exponential is where X is in our exponent or in a power of something. In this case, I cannot use K as the constant.
So let's look at quadratic first. Well, the first thing to understand is whenever we've got an equation, either a quadratic or cubic or could be a quartic, the term with the highest power, X, in this case, it's this term with the highest power of X. It's always going to be the term that kind of defines the general shape of the curve. So what we can really do to understand the general shape of a quadratic, first of all, is if I give myself some axes. Well, if X is equal to zero, x squared is going to be zero.
And if we times that by A, which is just some constant, some value, we get zero still, right? If X is one, well, one squared is going to be one times it by A, we get at that point. Now, if X is T, then we do two squared and we get four and we end up with A times four. So we end up with some deaths like that. Now, we can't imagine A is one for this to keep things simpler.
And we keep going with this, we'd get three squared as nine and we keep going. You basically end up with something that looks like this. And the same happens on the other side, right? Because the negative times the negative is a positive. So minus one times minus one will give us one.
Minus two times minus two will give us plus four, and minus three times minus three will give us plus nine and so on. So what we end up doing is we end up basically mirroring this curve around the Y axis. So that is the general shape of my ex squared. Now, if we were to introduce the BX term or the plus C terms, and what's going to happen is this is going to make me shift around a little bit. If we added a BX term into this well, on the right hand side, where X is positive, we're going to end up shifting everything up by a bit, aren't we?
Because we're going to add X or some multiple of X onto every point. And if we look at the negative side again, in this case, X is going to be negative. And so what we're going to be doing is we're going to be subtracting X from each point. And what we end up with is another graph that isn't going to look too dissimilar. They might just end up looking like that instead.
So that's our quadratics, right? You don't need to read too much into it. But the general shape is what's most important. With cubic, they're fairly similar in this case. Again, let's just focus on this term for now.
To keep things simple, let's just say a again is equal to one. So we're going to have zero cubed is just zero. One cube is just one. Two cubed is going to be eight. Three cubed is going to be we're already kind of shooting off the page, right on the negative side, the vectors minus one.
Well, we're going to have minus one times minus one to get plus one times another minus one, because we do minus one cubed. One minus one times minus one times minus one is going to give us minus one because minus times minus times minus gives us minus. And if we do minus two cubes, well, minus two times minus two is plus four. Plus four times minus two is going to be minus eight. And what we can see is we get this kind of weird sneaky shape.
Minus three cube is going to give us -27, down here somewhere. And so what we end up with is a shape that looks like that. That's our cubic shape. And again, if we were to add in these extra terms of A squared and X and a constant, all we're going to be doing is we're going to be changing the shape slightly. We might end up with a shape that could look like that, but it's going to be the same kind of shape, right?
It's going to go up and then plateau off, maybe go down and then go up again. It's going to be this kind of S shaped curve that we always get. Recip calls are fairly easy, quite nice compared to quadratics and cubic. So I give myself some axes. And in this case, let's just imagine if you values of X, right?
So if X is really, really tiny, well, some number, any number over a very tiny number is going to give us a massive number, right? If we were to divide five by one, we'd end up with something like 50,000. And again, let's just say K is one for now, to keep things simple, what we can say is next is getting smaller and smaller and smaller. This graph is going to be shooting up off the page. But if X is equal to one, or if X is equal to K, if we say K is just some unknown number at that point there, all we're doing is K is K.
Or we could be doing one over one if K is one. So one over one just gives us one. And so we pass through that point there and again as we continue on down. Well, as X gets much much bigger. Anything over a very large number gives us a very small number.
And so as X gets bigger and bigger and bigger al Qaeda and the value of each Y point gets smaller. And if we look at the negative side, well, if X is negative this whole thing is going to be the exact same copied over. But it's going to be negative because we're doing the same number as before, the same absolute positive number here divided by a negative number now. And so positive divided by negative give us negative and we end up with a curve that looks like that. Finally our exponential graph will go over a little bit onto here.
So again, if we have our axes well, let's look at what's going on here. So again, I'm going to keep things simple. Let's just say K is equal to one. It doesn't matter what K is equal to, it's just going to change the shape slightly.
If X is equal to zero, anything to the zero is just one, right? So when X is zero, anything to the zero is just one. So we always pass through one. If say X is equal to one. Now so rather than zero, we're going to say X is equal to one.
Well, anything to the one is just itself, right? So we can imagine that's just a K where it's just keep it as K. So anything to the one, it just itself. X equals one. Our graph is going to equal K.
That point there is going to be K. If X equals T, win it with K squared rather than K, we're going to K squared, which could be, I don't know. If K is T, we get two times two gives us four. If X is three, we're going to get K cubed because there's always K to the power of X or K cubed. Again, if K is equal to two, in this case it's going to be roughly eight.
Right. So we're already shooting off the page again, raining up up here. And if we look at down on the negative side of the X axis, if you look at say X is minus one or k to the minus one. If you remember power rules, k minus one is just equal to one over k to the one. This minus here just means one over.
So we're going to end up with one over k here. Cause anything for one is just one. So we get one over k or a k is two and then one over two which is a half.
If we do the same thing again, if we look at X is equal to when X is minus T, we get a quarter and we can start to see our nice exponential curve for me here, right? It's going to roughly look like that, maybe avoid that last slightly ugly bit. But that's our rough shape of an exponential. And so we have four main graphs and the general shapes that we'd expect to get from those, right? Let's look at an example question.
So in this case, we're given six curves and we need to match up the equation with the curve or the graph. So we want to match up the letter of the graph with the equation. Well, let's just start off with looking at graphite. Let's start off with this one. We know from what we just did that this is looking very similar to a reciprocal, right?
It's looking very similar to one over X or k over X. And so I can see down here we have K of x, right? In this case, k is four. But we've got that same equation in this case. Well, in this case we've got what looks like an exponential, right?
It tapers down when we're to the left of the central Yaxis and it dramatically increases when we're on the right. I know that C is going to be my exponential. In this case, exponential is going to be here, right? That's my two to be X term. So I know that C is going to be that one.
Let's move on to D next. So D looks a lot like that standard cubic kind of graph, right? It could be that or it could be that. Anything that has an S shape in it is generally going to be a cubic. I can just find my cubic in this case, it's here.
And I know that D is going to match up with that equation. Let's look at F next. So F is our standard kind of smiley face. We always expect this when we've got a quadratic, right? So that's my standard quadratic shape of X squared.
And I can shift this up and down and leave it around, but it's only going to look something like this. I know F is going to be my quadratic, which in this case is our third equation, right, that has my X squared terminate, making it the quadratic. So that is going to be F-L-T. We haven't actually covered, but not too tricky. This just involves knowing your sign from your course sign is going to be E.
Cause sign always starts at zero and so that's going to be my sign and cause always starts at one. And so I know is going to be custom.
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