Video: Finding the Vertex of a Quadratic and Determining Its Nature

Video Transcript

In this video, we’re gonna talk about a quick way to find the coordinates of a vertex of a quadratic and to work out whether it’s gonna be a maximum or a minimum point on the curve. If you’d 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. They’re either u-shaped and you might call them “downy uppies” because from left to right they go down then up or they’re n-shaped and you might call them “uppy downies” because from left to right they go up and down. In the first case, there’s a point at the bottom of that curve that’s got a lower 𝑦-coordinate than any other point on the curve, and we call that the minimum point. In the other case, there’s a point at the top of the curve that’s got a higher 𝑦-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 𝑦-coordinates like this is a vertex.

Although lots of people just call them turning points. Now it’s 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, there’s a simple rule you can follow to do this. If you arrange the equation in the 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 format, then just looking to see if the value of 𝑎 is positive or negative tells you whether it’s a positive happy smiley curve or a negative sad-faced curve. And if it’s a positive smiley happy curve, then the vertex will be at the bottom; it will be a minimum. And if it’s a sad negative curve, then the vertex will be at the top and will be a maximum.

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

But if you found out that the curve didn’t cut the 𝑥-axis anywhere, then that’s not really gonna help you to find the 𝑥-coordinate at the vertex. And if you’ve already got your equation in the completing the square or vertex form, then that makes it really easy to find the 𝑥- and 𝑦-coordinates of the vertex. You can easily identify the 𝑥-coordinate here from the bit that’s in the square, and the remaining bit tells you the 𝑦-coordinate.

But some equations are easier to get into the completing the square format than others, so that’s not always gonna be the most convenient method. So we’re 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 𝑦 is equal to 𝑥 squared something times 𝑥 squared plus something times 𝑥 plus something then The 𝑥-coordinate of its vertex is given by minus 𝑏 over two 𝑎.

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

Well 𝑏 was negative three and 𝑎 was one, so that’s 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, I’ve got three over two, one point five. And luckily, that tallies with what we see on the graph.

So now we know the 𝑥-coordinate of that minimum we can work out the 𝑦-coordinate by just plugging that value of 𝑥 back into the original equation. So wherever we see 𝑥 in the equation, we just replace it with this 𝑥-coordinate that we’ve just found. So it’s 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, we’ve got an 𝑥-coordinate of one point five and a 𝑦-coordinate of one point seven five. So we started off with 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 to help us work out what the 𝑎, 𝑏, and 𝑐 values were. We then were able to identify whether it was a maximum or a minimum by looking at whether 𝑎 was positive or negative. We then used the formula minus 𝑏 over two 𝑎 to work out the 𝑥-coordinate of that vertex. And we then used that resulting 𝑥-coordinate plugged into the original equation to work out what the corresponding 𝑦-coordinate was.

Let’s go through one more example and then we’ll give you a couple to do. Find the coordinates and the nature of the vertex of 𝑦 equals minus two 𝑥 squared plus four 𝑥 minus seven. So firstly, 𝑎 is negative two, 𝑏 is four, and 𝑐 is negative seven. If 𝑎 is negative, we’re talking about a negative sad curve, which means that the vertex is going to be a maximum. And we’re going to use the minus 𝑏 over two 𝑎 formula to work out the 𝑥-coordinate of that maximum. And since 𝑏 is four and 𝑎 is negative two, the maximum 𝑥-coordinate is the negative of four over two times negative two. So that’s gonna be one.

Now I’m plugging that value of 𝑥 back into our formula for 𝑦, to our equation. 𝑦, the maximum 𝑦-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 it’s at one, negative five.

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

Okay then, so let’s work out the 𝑎, 𝑏, and 𝑐 values for 𝑎. So 𝑎 is five, 𝑏 is negative three, and 𝑐 is negative one. So 𝑎 is five, which is positive and smiley and happy, which means that the vertex is gonna be down at the bottom of that and we’re gonna have a minimum. So the 𝑥-coordinate of that minimum is gonna be at negative 𝑏 over two 𝑎 So that’s the negative of negative three over two times five, which is three-tenths or nought point three. And the corresponding 𝑦-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, we’ve got to multiply out the parentheses there to get it into the 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 format. And when we do that, we’re left with 𝑦 is negative two 𝑥 squared plus one 𝑥 plus six. And this means that 𝑎 is negative two, 𝑏 is one, and 𝑐 is six. So 𝑎 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 we’re gonna have a maximum. And to work out the coordinates of that vertex, again we’re gonna use the formula minus 𝑏 over two 𝑎 for 𝑥. And 𝑏 is one and 𝑎 is negative two, so that’s minus one over two times negative two, which comes out to be positive a quarter or nought point two five. Now we’re gonna plug that value into the original equation to work out what the corresponding 𝑦-coordinate is.

So that’s 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, we’ve worked out that the vertex is a maximum and it’s at nought point two five, six point one two five.

So just summarising those steps then, first you’ve gotta get that quadratic into the form 𝑦 is equal to 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. Then if 𝑎 is greater than zero, it’s positive, so it’s a positive happy smiley curve. So that point, the turning point, the vertex is gonna be the bottom of the curve, so we’re gonna have a minimum. And if 𝑎 is less than zero, it’s negative and sad, which means that the vertex, the turning point is gonna be the top of the curve, which means we’re gonna have a maximum. To find the 𝑥-coordinate of the vertex, use 𝑥 is equal to negative 𝑏 over two 𝑎. And then finally substitute that 𝑥-value back into the original equation to find the corresponding 𝑦-coordinate of the vertex.