### Video Transcript

In this video, weโre gonna talk about a quick way to find the coordinates of a vertex of a quadratic and to work out whether itโs gonna be a maximum or a minimum point on the curve. If youโd like to explore some other ways of working out these things or understand where the formula comes from, then check out our video exploring different ways of locating vertices of quadratics.

First, all quadratics are symmetrical parabolas. Theyโre either u-shaped and you might call them โdowny uppiesโ because from left to right they go down then up or theyโre n-shaped and you might call them โuppy downiesโ because from left to right they go up and down. In the first case, thereโs a point at the bottom of that curve thatโs got a lower ๐ฆ-coordinate than any other point on the curve, and we call that the minimum point. In the other case, thereโs a point at the top of the curve thatโs got a higher ๐ฆ-coordinate than any other point on the curve, and we call that the maximum point. The general term for a point on the curve that marks the turning point between increasing and decreasing ๐ฆ-coordinates like this is a vertex.

Although lots of people just call them turning points. Now itโs a really useful skill to be able to take a look at the equation of a quadratic and to be instantly able to tell if the vertex is gonna be a minimum point or a maximum point. Now luckily, thereโs a simple rule you can follow to do this. If you arrange the equation in the ๐ฆ equals ๐๐ฅ squared plus ๐๐ฅ plus ๐ format, then just looking to see if the value of ๐ is positive or negative tells you whether itโs a positive happy smiley curve or a negative sad-faced curve. And if itโs a positive smiley happy curve, then the vertex will be at the bottom; it will be a minimum. And if itโs a sad negative curve, then the vertex will be at the top and will be a maximum.

So thatโs pretty easy. All we need to do now is to find out how to work out the ๐ฅ- and ๐ฆ-coordinates of that point and weโre away. Well, there several ways to do this. For example, if youโve already worked out where the curve cuts the ๐ฅ-axis, because the parabola is symmetrical if you think about the midpoint between those two points there, that will tell you the ๐ฅ-coordinate of the vertex. And if you worked that out and it only touched the ๐ฅ-axis in one place, then you know youโve got the vertex there. So youโll know the ๐ฅ-coordinate of the vertex.

But if you found out that the curve didnโt cut the ๐ฅ-axis anywhere, then thatโs not really gonna help you to find the ๐ฅ-coordinate at the vertex. And if youโve already got your equation in the completing the square or vertex form, then that makes it really easy to find the ๐ฅ- and ๐ฆ-coordinates of the vertex. You can easily identify the ๐ฅ-coordinate here from the bit thatโs in the square, and the remaining bit tells you the ๐ฆ-coordinate.

But some equations are easier to get into the completing the square format than others, so thatโs not always gonna be the most convenient method. So weโre just gonna learn about a simple but very effective method, and all you have to do is remember one simple formula. If a quadratic is in the form ๐ฆ is equal to ๐ฅ squared something times ๐ฅ squared plus something times ๐ฅ plus something then The ๐ฅ-coordinate of its vertex is given by minus ๐ over two ๐.

For example, hereโs a quadratic ๐ฆ equals ๐ฅ squared minus three ๐ฅ plus four. So in that format, we can tell that ๐ is one because itโs one ๐ฅ squared, ๐ is negative three cause itโs minus three ๐ฅ, and ๐ is four because itโs positive four. The value of ๐, one is positive so we know itโs gonna be a happy curve and we can see that from the graph anyway. But if itโs a happy curve, the turning point, the vertex, is gonna be at the bottom; itโs gonna be a minimum. And the ๐ฅ-coordinate of that minimum is gonna be at minus ๐ over two ๐.

Well ๐ was negative three and ๐ was one, so thatโs gonna be the negative of negative three over two times one, which when I sort out the negative signs and sort it and cancel it down, Iโve got three over two, one point five. And luckily, that tallies with what we see on the graph.

So now we know the ๐ฅ-coordinate of that minimum we can work out the ๐ฆ-coordinate by just plugging that value of ๐ฅ back into the original equation. So wherever we see ๐ฅ in the equation, we just replace it with this ๐ฅ-coordinate that weโve just found. So itโs gonna be one times three over two squared minus three times three over two plus four. And when we work that lot out, we get seven over four.

Or if you want to convert that to decimals, weโve got an ๐ฅ-coordinate of one point five and a ๐ฆ-coordinate of one point seven five. So we started off with ๐ฆ equals ๐๐ฅ squared plus ๐๐ฅ plus ๐ to help us work out what the ๐, ๐, and ๐ values were. We then were able to identify whether it was a maximum or a minimum by looking at whether ๐ was positive or negative. We then used the formula minus ๐ over two ๐ to work out the ๐ฅ-coordinate of that vertex. And we then used that resulting ๐ฅ-coordinate plugged into the original equation to work out what the corresponding ๐ฆ-coordinate was.

Letโs go through one more example and then weโll give you a couple to do. Find the coordinates and the nature of the vertex of ๐ฆ equals minus two ๐ฅ squared plus four ๐ฅ minus seven. So firstly, ๐ is negative two, ๐ is four, and ๐ is negative seven. If ๐ is negative, weโre talking about a negative sad curve, which means that the vertex is going to be a maximum. And weโre going to use the minus ๐ over two ๐ formula to work out the ๐ฅ-coordinate of that maximum. And since ๐ is four and ๐ is negative two, the maximum ๐ฅ-coordinate is the negative of four over two times negative two. So thatโs gonna be one.

Now Iโm plugging that value of ๐ฅ back into our formula for ๐ฆ, to our equation. ๐ฆ, the maximum ๐ฆ-coordinate is gonna be negative two times one squared plus four times one minus seven, which works out to be negative five. And the answer to the question then is that the vertex is a maximum and itโs at one, negative five.

Okay, now your turn. So read these and I want you to pause the video and then come back, so Iโm gonna wait three seconds and give you the answer. So youโve got to find the nature and coordinates of the vertices of ๐ฆ equals five ๐ฅ squared minus three ๐ฅ minus one and ๐ฆ equals two ๐ฅ plus three times two minus ๐ฅ.

Okay then, so letโs work out the ๐, ๐, and ๐ values for ๐. So ๐ is five, ๐ is negative three, and ๐ is negative one. So ๐ is five, which is positive and smiley and happy, which means that the vertex is gonna be down at the bottom of that and weโre gonna have a minimum. So the ๐ฅ-coordinate of that minimum is gonna be at negative ๐ over two ๐ So thatโs the negative of negative three over two times five, which is three-tenths or nought point three. And the corresponding ๐ฆ-coordinate for that minimum point is gonna be five times nought point three squared minus three times nought point three minus one, which works out to be negative one point four five. So in the first case, the vertex is a minimum at nought point three, negative one point four five.

Now for the second one, weโve got to multiply out the parentheses there to get it into the ๐๐ฅ squared plus ๐๐ฅ plus ๐ format. And when we do that, weโre left with ๐ฆ is negative two ๐ฅ squared plus one ๐ฅ plus six. And this means that ๐ is negative two, ๐ is one, and ๐ is six. So ๐ is negative and sad. It looks like a sad mouth, which means that the vertex is gonna be at the top of that curve, so weโre gonna have a maximum. And to work out the coordinates of that vertex, again weโre gonna use the formula minus ๐ over two ๐ for ๐ฅ. And ๐ is one and ๐ is negative two, so thatโs minus one over two times negative two, which comes out to be positive a quarter or nought point two five. Now weโre gonna plug that value into the original equation to work out what the corresponding ๐ฆ-coordinate is.

So thatโs gonna be negative two times nought point two five squared plus nought point two five plus six, and that equals six and an eighth which is six point one two five. So for the second question, weโve worked out that the vertex is a maximum and itโs at nought point two five, six point one two five.

So just summarising those steps then, first youโve gotta get that quadratic into the form ๐ฆ is equal to ๐๐ฅ squared plus ๐๐ฅ plus ๐. Then if ๐ is greater than zero, itโs positive, so itโs a positive happy smiley curve. So that point, the turning point, the vertex is gonna be the bottom of the curve, so weโre gonna have a minimum. And if ๐ is less than zero, itโs negative and sad, which means that the vertex, the turning point is gonna be the top of the curve, which means weโre gonna have a maximum. To find the ๐ฅ-coordinate of the vertex, use ๐ฅ is equal to negative ๐ over two ๐. And then finally substitute that ๐ฅ-value back into the original equation to find the corresponding ๐ฆ-coordinate of the vertex.