Video: Introducing Quadratic Equations and Matching Equations to Their Graphs

An introduction to quadratic equations and their graphs. We explore the general format of the equation π¦ = ππ₯Β² + ππ₯ + π and the effects of changing the values of π, π, and π on the shape, position, and orientation of the graph.

18:00

Video Transcript

Weβre going to talk about quadratic equations. This is a family of equations which have lots of uses in the real world. And weβre going to look at the format of the equation and some graphs. And weβ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 π₯ squared term, an π₯ term, and a constant. And some people call them a βpolynomial of order twoβ because the highest power or exponent of π₯ is two π₯ squared.

Okay so you get the message. Theyβre really useful and theyβre really important. So letβ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 theyβre symmetrical curves which either point upwards or downwards, but they extend out to infinity either way in the π₯-direction. So they go down to minus infinity and up to plus infinity. Now thatβs quite an important point to remember because it might not look like whatβs going to happen when youβre actually plotting these things. So because youβre taking your π₯-coordinate and youβre squaring it, it can get very, very big. So your π¦-coordinate increases much more quickly than your π₯-coordinate does.

This means if your π₯- and π¦-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 π₯-direction or negative infinity in the π₯-direction, but believe me it really is. We say that the general form of the quadratic equation is π¦ equals ππ₯ squared plus ππ₯ plus π. Now the π, the π, and the π are just numbers and changing the values of π, π, and π will affect the shape, the orientation, and the position of the parabola on a graph. So it just means, you know, π¦ equals some number times π₯ squared plus a different number times π₯ plus a different number on its own. So for example, π¦ equals five π₯ squared minus two π₯ plus seven is a particular quadratic equation. And relating that back to our general form, π would be equal to five in this case, the π would be equal to negative two, and the π would be equal to seven.

Another quadratic equation could be π¦ equals minus three π₯ squared plus nine. And in this case π would be negative three, the π would be zero β because weβve got no π₯ terms; so itβs zero times π₯, but we donβt normally bother actually writing that β and the π would be equal to nine. If youβve got access to some graphing software, itβ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 π and π and π and see what effect they have on the shape of the curve. Come back here when youβre finished and Iβm going to talk you through what I think you should have found.

So Iβm going to set π and π to be zero. And Iβm just going to play around with different values of π to see what effect they have on the curve. First one then π is nought point one and π is zero and π is zero. This is what the curve looks like. Well I set π to nought point five. This is what the curve looks like. Now on this scale it doesnβt look like that curve is quite gonna go up to plus infinity. But look I can put any value that I like for π₯ into this equation and then I square it and then halve that value to get the π¦-coordinate. So there must be points on the curve with π₯-coordinates going off to plus infinity or minus infinity. But theyβ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 π is one. This is what π¦ equals one π₯ squared looks like. When π equals two, π¦ equals two π₯ squared. And this is what the curve looks like. When π equals five, π¦ is equal to five π₯ squared. And thatβs what the curve looks like. And when π equals ten, π¦ is equal to ten π₯ squared. And thatβs what the curve looks like. So what you can see is that as π increases, we are multiplying the π¦-coordinate by a bigger and bigger number. Weβ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 π¦-axis.

But look what happens when we make π negative. So weβve got π is negative nought point one. So weβre looking at the curve π¦ equals negative nought point one π₯ squared. So itβs gone back to this sort of fatter curve. Itβs been squashed up the π¦-axis there. And- but now itβs gone down below the π₯-axis. It looks like this is a sad face rather than that kind of happy smiley face that we saw before. Now π is negative nought point five. Weβve got π¦ equals negative nought point five π₯ squared. Itβs still like a sad face down there and itβs got a bit thinner again. Now weβve got π is equal to negative one. So π¦ is equal to negative one π₯ squared or just negative π₯ squared. So again the curve got a bit thinner, but still it looks like the sad face. When π is negative two, it gets thinner still, π is negative five, it gets thinner still, and when π is negative ten, itβs thinner still.

So bring a few graphs together on the same page. When π is positive, we get these smiley face graphs. And you know when youβre positive, youβre happy; you make this smiley face. So that kinda make sense. And as π gets further away from zero, it gets more and more positive. The graph gets thinner and thinner as we stretch it up the π¦-axis. And looking at these negative values of π, we can say when youβre negative youβ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 π gets away from zero β so negative ten is a large negative number β the thinner that graph gets as we stretch along the π¦-axis. So the sign of the π₯ 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 itβs just close to zero, itβs going to be one of these wide curves here; if itβs a larger value, then itβs going to be a thin curve because itβs stretched up the π¦-axis.

Right, now weβre gonna mess around with the value of π. So weβre gonna start off with this: π¦ is equal to nought point two π₯ squared plus zero. So π is zero, π is zero, and π is nought point two. Now weβve kept everything the same except weβve increased π to be one. So weβve got π¦ equals nought point two π₯ squared plus one. Thatβs what the graph looks like. Increasing π to two gives us this and increasing π to three gives us this. Now the only thing that is different about these curves is where it cuts the π¦-axis. When π is zero, it cuts the π¦-axis at zero, when π is one, it cuts the π¦-axis at one, when π is two, it cuts the π¦-axis at two, and when π is three, it cuts the π¦-axis at three. Now that makes sense because if you think about it on the π¦-axis, all of the coordinates have got an π₯-coordinate of zero.

And when we put an π₯-coordinate of zero into our equation, zero squared is zero, so weβve got π times zero which is zero, weβve got π times zero which is zero. So on the π¦-axis when π₯ is equal to zero, the π¦-coordinate is just gonna be the same as whatever that π-value was, the constant on the end of our quadratic equation. And just to confirm that. Weβve done the same with π is negative one, negative two, and negative three. When π is negative one, it cuts the π¦-axis at negative one, and π is negative two, it cuts the π¦-axis at negative two, and when π is negative three, it cuts the π¦-axis at negative three. So the constant term on the end of our quadratic simply tells us where does that curve cut through the π¦-axis.

Letβs have a quick look at what happens when we put mess around with the π parameter then. So setting π equal to three gives us π¦ equals nought point two π₯ squared plus three π₯. Setting π equal to two gives us π¦ equals nought point two π₯ squared plus two π₯. Setting π equal to one gives us π¦ equals nought point two π₯ squared plus one π₯. And setting π equal to zero gives us π¦ equals nought point two π₯ squared plus zero π₯ or just π¦ equals nought point two π₯ squared. So what we can see is that when the π-value is larger and positive, itβs moved the whole curve to the left. Because π was zero in every case, the curve still cuts the π¦-axis at zero in the same place. But when π is bigger and positive, itβs moved the curve to the left. π is positive, but not quite as big. Itβs not quite as far to the left. π still positive, itβs moved a little bit to the left. And when π is zero, itβs symmetrical about the π¦-axis. Itβs needed to go on to the left, not the right of the π¦-axis.

So starting off with π is equal to zero, weβll then check out some negative values of π. So when π is negative one, weβve got π¦ is nought point two π₯ squared minus one π₯ and the graph looks like that. When π is negative two, we end up with π¦ equals nought point two π₯ squared minus two π₯ and the graph looks like that. And when π is negative three, weβve got π¦ equals nought point two π₯ squared minus three π₯ and the graph looks like that, even further to the right. So in this case having a negative π-value shifts the graph over to the right. Having π equal to zero, remember weβve got our graph, our parabola, which is symmetric about the π¦-axis. So the π parameters seem to move the graph left or right. But notice because π is zero in every case the curve still cuts the π¦-axis at zero. So thatβs remains unchanged. Although weβre moving it to left and right, weβre still keeping it cutting the π¦-axis at zero. Now we do need to be a little bit careful with this thing about π being moved left and right. Because when π is negative and weβve got a sad curve, then making π more negative moves it further to the left rather than to the right. So π is negative two, negative nought point two. Again π is negative two; itβs moved to the left but not quite as much. And when π is negative nought point two and π is negative one, itβs still moved to the left slightly, but not as much again. But again because in all cases π was equal to zero, the curve did cut the π¦-axis at zero.

So just to summarise that then. The π parameter moves the curve to the left or right, but it does kinda of depend on the value of π. So when π is positive as we have here, a positive value of π will move the curve to the left of the π¦-axis. When π is negative, a positive value of π moves the curve to the right side of the axis. And now when π is negative if π was positive, then a negative π moves the curve to the right of the π¦-axis. And if π was negative, then a negative π moves the curve to the left of the π¦-axis. But the thing to know there is changing the value of π doesnβt make the curve wider or thinner; it just moves it left or right and it doesnβt affect where it cuts the π¦-axis.

So now letβ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 π¦-axis at zero, so that means that π is equal to zero. So just filling in the π- and π-values on those equations, we can see that equation three canβt be right because the π-value here is three. Now we also noticed that itβs a positive smiley happy curve. So that means that π is positive. So π is bigger than zero. And that could match either the first or the second equation, where π is equal to one; thatβs positive. Now the next thing we noticed is that the curve is symmetrical about the π¦-axis. So it hasnβt been moved to the left and hasnβt been moved to the right. That must mean that π is equal to zero. And in the second equation, π was equal to two. So it canβt be that one either, so our answer is π¦ equals π₯ squared.

Next question then: which of these equations matches this graph? So first Iβm just going to fill in the π-, π-, and π-values for each of those. So in the first one youβll see π is three, π is zero, π is zero; the second one π is two, π is negative three, and π is zero, and on the third one π is one, π is negative two, and π is three. So looking at the graph, we can see that it cuts the π¦-axis at zero. This means that π must be equal to zero. So again thatβs ruling out our third equation, which has π equal to three. So now we can see that the curve isnβt symmetrical about the π¦-axis; itβs been shifted off to the right. So this means that π isnβt zero. Now if you donβt remember which way around; is it moved to left? Is it to the right? donβt worry about that too much because in this quite particular question here weβve got a choice between two and then the first one. The π-value is zero, so we know that thatβs not correct. But if you do remember- remember when π is positive and negative π value moves it to the right and a positive π-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 π-, π-, and π-values and the first thing to notice is that it cuts the π¦-axis at three. So π is equal to three. This rules out our second equation which has π is equal to negative three. We see that it is a negative sad curve. So π is less than zero. Well that matches both of the remaining equations. But itβs not symmetrical about the π¦-axis. Itβs been shifted to the right. So π canβt be equal to zero. So that rules out number one. So our equation is π¦ equals negative nought point five π₯ squared plus two π₯ plus three.