We’re going to talk about quadratic equations. This is a family of equations which have lots of uses in the real world. And we’re going to look at the format of the equation and some graphs. And we’re gonna learn about how we can change different aspects of the equation to have different effects on the graph. So first what is a quadratic equation? Well a quadratic equation is an equation with an 𝑥 squared term, an 𝑥 term, and a constant. And some people call them a “polynomial of order two” because the highest power or exponent of 𝑥 is two 𝑥 squared.
Now it could be the case that the coefficient of the 𝑥 term or the constant term or even both of them are zero. And these would still count as quadratic equations. So 𝑦 equals two 𝑥 squared, 𝑦 equals 𝑥 squared plus five, 𝑦 equals four 𝑥 squared minus 𝑥 are all examples of quadratic equations. And it’s also okay for the letters to be something other than 𝑥 or 𝑦. So we’ve got here 𝑠 equals 𝑢𝑡 plus half 𝑎𝑡 squared; that’s a quadratic in 𝑡. 𝑒 equals a half 𝑚𝑣 squared; that’s a quadratic in 𝑣. But you can’t have any terms which have got a higher exponent or power than two, so no 𝑥 cubed terms for example, no fractional exponents, no negative exponents; just a constant number, a multiple of 𝑥, and a multiple of 𝑥 squared. So these two equations here are not quadratics. This’s got an 𝑥 squared term, an 𝑥 term, and a constant; so that’s good. But then it’s got one over 𝑥, so that is not a quadratic. This one here has got an 𝑥 squared term, has got an 𝑥 term, but- and it’s got a number, but it’s got this square root of 𝑥. So that again is not a quadratic. It can only have these three things: an 𝑥 squared term, an 𝑥 term, and a constant. Now we know what a quadratic equation is. Why are they called quadratics? You’d think with the term quad in there, it would have exponents of four. But in fact it’s based on the Latin word “quadratum” which means square. To work out the area of a square, you multiply the length of the side by itself; so you get a squared term. What do you think of that? It’s still easier than saying polynomial of order two every time; so quadratics it is.
So we’ve got quadratic equations which have up to squared terms, but what are they used for? Well when you throw a ball if we ignore friction and air resistance, a path that they follow through the air is roughly a quadratic. If we were to try to analyse an asteroid hurtling towards the Earth, we would use quadratic equations to model that situation to make our predictions about where and when it was going to hit the ground. Working out the length of a pendulum needs to be if it’s going to take a certain amount of time to swing backwards and forwards uses a quadratic formula. The shapes of mirrors or dishes on reflecting telescopes follow quadratic curves. The math that we use to work out how fast you’re going at the start of a skid before an accident involves quadratic equations. To calculate the net resistance of a bunch of resistors in parallel electric circuit also needs quadratic equations. Working out the optimal size for packaging often requires quadratics. You can even use quadratics to model the fly path of bees going to from their hive.
Okay so you get the message. They’re really useful and they’re really important. So let’s find out a bit more about the equations and what their graphs look like. So all quadratics have graphs which are parabola-shaped like these. So they’re symmetrical curves which either point upwards or downwards, but they extend out to infinity either way in the 𝑥-direction. So they go down to minus infinity and up to plus infinity. Now that’s quite an important point to remember because it might not look like what’s going to happen when you’re actually plotting these things. So because you’re taking your 𝑥-coordinate and you’re squaring it, it can get very, very big. So your 𝑦-coordinate increases much more quickly than your 𝑥-coordinate does.
This means if your 𝑥- and 𝑦-axes are based on the same scale, so one unit in this direction is the same as one unit in this direction. It might look as though your parabola is so thin, is never gonna go to positive infinity in the 𝑥-direction or negative infinity in the 𝑥-direction, but believe me it really is. We say that the general form of the quadratic equation is 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. Now the 𝑎, the 𝑏, and the 𝑐 are just numbers and changing the values of 𝑎, 𝑏, and 𝑐 will affect the shape, the orientation, and the position of the parabola on a graph. So it just means, you know, 𝑦 equals some number times 𝑥 squared plus a different number times 𝑥 plus a different number on its own. So for example, 𝑦 equals five 𝑥 squared minus two 𝑥 plus seven is a particular quadratic equation. And relating that back to our general form, 𝑎 would be equal to five in this case, the 𝑏 would be equal to negative two, and the 𝑐 would be equal to seven.
Another quadratic equation could be 𝑦 equals minus three 𝑥 squared plus nine. And in this case 𝑎 would be negative three, the 𝑏 would be zero — because we’ve got no 𝑥 terms; so it’s zero times 𝑥, but we don’t normally bother actually writing that — and the 𝑐 would be equal to nine. If you’ve got access to some graphing software, it’ll be a brilliant idea to go to pause the video now and go and try using the software and getting into plots and quadratic curves. Try different values for 𝑎 and 𝑏 and 𝑐 and see what effect they have on the shape of the curve. Come back here when you’re finished and I’m going to talk you through what I think you should have found.
So I’m going to set 𝑏 and 𝑐 to be zero. And I’m just going to play around with different values of 𝑎 to see what effect they have on the curve. First one then 𝑎 is nought point one and 𝑏 is zero and 𝑐 is zero. This is what the curve looks like. Well I set 𝑎 to nought point five. This is what the curve looks like. Now on this scale it doesn’t look like that curve is quite gonna go up to plus infinity. But look I can put any value that I like for 𝑥 into this equation and then I square it and then halve that value to get the 𝑦-coordinate. So there must be points on the curve with 𝑥-coordinates going off to plus infinity or minus infinity. But they’re gonna be way off the top of the page up there, so we have to zoom out a long, long way to see where those points are.
So now 𝑎 is one. This is what 𝑦 equals one 𝑥 squared looks like. When 𝑎 equals two, 𝑦 equals two 𝑥 squared. And this is what the curve looks like. When 𝑎 equals five, 𝑦 is equal to five 𝑥 squared. And that’s what the curve looks like. And when 𝑎 equals ten, 𝑦 is equal to ten 𝑥 squared. And that’s what the curve looks like. So what you can see is that as 𝑎 increases, we are multiplying the 𝑦-coordinate by a bigger and bigger number. We’re stretching that curve up in that direction, in the vertical direction. So the curve appears to get kinda of thinner and stretched up the 𝑦-axis.
But look what happens when we make 𝑎 negative. So we’ve got 𝑎 is negative nought point one. So we’re looking at the curve 𝑦 equals negative nought point one 𝑥 squared. So it’s gone back to this sort of fatter curve. It’s been squashed up the 𝑦-axis there. And- but now it’s gone down below the 𝑥-axis. It looks like this is a sad face rather than that kind of happy smiley face that we saw before. Now 𝑎 is negative nought point five. We’ve got 𝑦 equals negative nought point five 𝑥 squared. It’s still like a sad face down there and it’s got a bit thinner again. Now we’ve got 𝑎 is equal to negative one. So 𝑦 is equal to negative one 𝑥 squared or just negative 𝑥 squared. So again the curve got a bit thinner, but still it looks like the sad face. When 𝑎 is negative two, it gets thinner still, 𝑎 is negative five, it gets thinner still, and when 𝑎 is negative ten, it’s thinner still.
So bring a few graphs together on the same page. When 𝑎 is positive, we get these smiley face graphs. And you know when you’re positive, you’re happy; you make this smiley face. So that kinda make sense. And as 𝑎 gets further away from zero, it gets more and more positive. The graph gets thinner and thinner as we stretch it up the 𝑦-axis. And looking at these negative values of 𝑎, we can say when you’re negative you’re a bit sad; you get a sad face. Now so that kinda make sense that we get these sad face curves. And the further 𝑎 gets away from zero — so negative ten is a large negative number — the thinner that graph gets as we stretch along the 𝑦-axis. So the sign of the 𝑥 squared coefficient tells us whether it is going to be a happy curve or a sad curve. And the value of it tells us how thin the curve is going to be: if it’s just close to zero, it’s going to be one of these wide curves here; if it’s a larger value, then it’s going to be a thin curve because it’s stretched up the 𝑦-axis.
Right, now we’re gonna mess around with the value of 𝑐. So we’re gonna start off with this: 𝑦 is equal to nought point two 𝑥 squared plus zero. So 𝑐 is zero, 𝑏 is zero, and 𝑎 is nought point two. Now we’ve kept everything the same except we’ve increased 𝑐 to be one. So we’ve got 𝑦 equals nought point two 𝑥 squared plus one. That’s what the graph looks like. Increasing 𝑐 to two gives us this and increasing 𝑐 to three gives us this. Now the only thing that is different about these curves is where it cuts the 𝑦-axis. When 𝑐 is zero, it cuts the 𝑦-axis at zero, when 𝑐 is one, it cuts the 𝑦-axis at one, when 𝑐 is two, it cuts the 𝑦-axis at two, and when 𝑐 is three, it cuts the 𝑦-axis at three. Now that makes sense because if you think about it on the 𝑦-axis, all of the coordinates have got an 𝑥-coordinate of zero.
And when we put an 𝑥-coordinate of zero into our equation, zero squared is zero, so we’ve got 𝑎 times zero which is zero, we’ve got 𝑏 times zero which is zero. So on the 𝑦-axis when 𝑥 is equal to zero, the 𝑦-coordinate is just gonna be the same as whatever that 𝑐-value was, the constant on the end of our quadratic equation. And just to confirm that. We’ve done the same with 𝑐 is negative one, negative two, and negative three. When 𝑐 is negative one, it cuts the 𝑦-axis at negative one, and 𝑐 is negative two, it cuts the 𝑦-axis at negative two, and when 𝑐 is negative three, it cuts the 𝑦-axis at negative three. So the constant term on the end of our quadratic simply tells us where does that curve cut through the 𝑦-axis.
Let’s have a quick look at what happens when we put mess around with the 𝑏 parameter then. So setting 𝑏 equal to three gives us 𝑦 equals nought point two 𝑥 squared plus three 𝑥. Setting 𝑏 equal to two gives us 𝑦 equals nought point two 𝑥 squared plus two 𝑥. Setting 𝑏 equal to one gives us 𝑦 equals nought point two 𝑥 squared plus one 𝑥. And setting 𝑏 equal to zero gives us 𝑦 equals nought point two 𝑥 squared plus zero 𝑥 or just 𝑦 equals nought point two 𝑥 squared. So what we can see is that when the 𝑏-value is larger and positive, it’s moved the whole curve to the left. Because 𝑐 was zero in every case, the curve still cuts the 𝑦-axis at zero in the same place. But when 𝑏 is bigger and positive, it’s moved the curve to the left. 𝑏 is positive, but not quite as big. It’s not quite as far to the left. 𝑏 still positive, it’s moved a little bit to the left. And when 𝑏 is zero, it’s symmetrical about the 𝑦-axis. It’s needed to go on to the left, not the right of the 𝑦-axis.
So starting off with 𝑏 is equal to zero, we’ll then check out some negative values of 𝑏. So when 𝑏 is negative one, we’ve got 𝑦 is nought point two 𝑥 squared minus one 𝑥 and the graph looks like that. When 𝑏 is negative two, we end up with 𝑦 equals nought point two 𝑥 squared minus two 𝑥 and the graph looks like that. And when 𝑏 is negative three, we’ve got 𝑦 equals nought point two 𝑥 squared minus three 𝑥 and the graph looks like that, even further to the right. So in this case having a negative 𝑏-value shifts the graph over to the right. Having 𝑏 equal to zero, remember we’ve got our graph, our parabola, which is symmetric about the 𝑦-axis. So the 𝑏 parameters seem to move the graph left or right. But notice because 𝑐 is zero in every case the curve still cuts the 𝑦-axis at zero. So that’s remains unchanged. Although we’re moving it to left and right, we’re still keeping it cutting the 𝑦-axis at zero. Now we do need to be a little bit careful with this thing about 𝑏 being moved left and right. Because when 𝑎 is negative and we’ve got a sad curve, then making 𝑏 more negative moves it further to the left rather than to the right. So 𝑎 is negative two, negative nought point two. Again 𝑏 is negative two; it’s moved to the left but not quite as much. And when 𝑎 is negative nought point two and 𝑏 is negative one, it’s still moved to the left slightly, but not as much again. But again because in all cases 𝑐 was equal to zero, the curve did cut the 𝑦-axis at zero.
So just to summarise that then. The 𝑏 parameter moves the curve to the left or right, but it does kinda of depend on the value of 𝑎. So when 𝑎 is positive as we have here, a positive value of 𝑏 will move the curve to the left of the 𝑦-axis. When 𝑎 is negative, a positive value of 𝑏 moves the curve to the right side of the axis. And now when 𝑏 is negative if 𝑎 was positive, then a negative 𝑏 moves the curve to the right of the 𝑦-axis. And if 𝑎 was negative, then a negative 𝑏 moves the curve to the left of the 𝑦-axis. But the thing to know there is changing the value of 𝑏 doesn’t make the curve wider or thinner; it just moves it left or right and it doesn’t affect where it cuts the 𝑦-axis.
So now let’s have a go matching some graphs to their equations based on what we know. Question one: which of these curves matches the graph? Well the curve cuts the 𝑦-axis at zero, so that means that 𝑐 is equal to zero. So just filling in the 𝑏- and 𝑐-values on those equations, we can see that equation three can’t be right because the 𝑐-value here is three. Now we also noticed that it’s a positive smiley happy curve. So that means that 𝑎 is positive. So 𝑎 is bigger than zero. And that could match either the first or the second equation, where 𝑎 is equal to one; that’s positive. Now the next thing we noticed is that the curve is symmetrical about the 𝑦-axis. So it hasn’t been moved to the left and hasn’t been moved to the right. That must mean that 𝑏 is equal to zero. And in the second equation, 𝑏 was equal to two. So it can’t be that one either, so our answer is 𝑦 equals 𝑥 squared.
Next question then: which of these equations matches this graph? So first I’m just going to fill in the 𝑎-, 𝑏-, and 𝑐-values for each of those. So in the first one you’ll see 𝑎 is three, 𝑏 is zero, 𝑐 is zero; the second one 𝑎 is two, 𝑏 is negative three, and 𝑐 is zero, and on the third one 𝑎 is one, 𝑏 is negative two, and 𝑐 is three. So looking at the graph, we can see that it cuts the 𝑦-axis at zero. This means that 𝑐 must be equal to zero. So again that’s ruling out our third equation, which has 𝑐 equal to three. So now we can see that the curve isn’t symmetrical about the 𝑦-axis; it’s been shifted off to the right. So this means that 𝑏 isn’t zero. Now if you don’t remember which way around; is it moved to left? Is it to the right? don’t worry about that too much because in this quite particular question here we’ve got a choice between two and then the first one. The 𝑏-value is zero, so we know that that’s not correct. But if you do remember- remember when 𝑎 is positive and negative 𝑏 value moves it to the right and a positive 𝑏-value would move it to the left, so this is our equation.
So pause the video now and have a go at this question. Well filling in the 𝑎-, 𝑏-, and 𝑐-values and the first thing to notice is that it cuts the 𝑦-axis at three. So 𝑐 is equal to three. This rules out our second equation which has 𝑐 is equal to negative three. We see that it is a negative sad curve. So 𝑎 is less than zero. Well that matches both of the remaining equations. But it’s not symmetrical about the 𝑦-axis. It’s been shifted to the right. So 𝑏 can’t be equal to zero. So that rules out number one. So our equation is 𝑦 equals negative nought point five 𝑥 squared plus two 𝑥 plus three.