Hi, and welcome. My name is Anthony Varela. And today, we're going to convert a quadratic equation into a vertex form.
So we're going to talk a bit about vertex form of quadratic equations. We're going to talk about how to convert it from an equation written in standard form. And this is going to involve a process called completing the square.
So what is the vertex form of a quadratic equation? Well, that's y equals a times x minus h quantity squared plus k. Now we call it vertex form, because h and k form the x and y-coordinates of the parabolas vertex.
So here is a look at what a vertex looks like when graphed. When we have a u shaped parabola that opens upwards, it's a minimum point. If this were a parabola that open downwards, it would be a maximum point. And it lies on the axis of symmetry.
So the axis of symmetry is a line of reflection. So if we were to plot a couple of points on one side of this vertex, we could then just reflect it across this line of reflection and get more points on the other side of that axis of symmetry. So that's why we would want to convert an equation from standard form into vertex form, because we can easily identify the vertex that would also help us easily identify that axis of symmetry.
So we can plot points using reflection. So we're going to be converting from standard forms. So let's compare these two forms for a second. Standard form is y equals ax squared plus bx plus c. And we can see that our terms are written out expanded.
And in vertex form, part of the equation is a binomial squared x minus h quantity squared. Now we can write a binomial squared from what's called a perfect square trinomial. And here's an example of that, x squared plus 8x plus 16.
This is called a perfect square trinomial because we see three terms, but it can be written as a binomial squared, x plus 4 quantity squared. And notice if we take this 4 and double that, we get the x term coefficient. And if we square it, we get this constant term.
So as we're converting, we're going to want to look for something like this. That could be then written as a binomial squared. So this involves then completing the square. So we'll see how this looks in our example. So here's our example. y equals 2x squared plus 12x plus 16.
Now, the first step is we're going to move that constant term over to the other side of the equation. So we see plus 16. So we're going to subtract 16 from both sides. So what we end up with now is y minus 16 equals 2x squared plus 12x.
So now that we don't have that constant term what we're going to do to that side of the equation is going to take a look at the coefficient of the x squared term and factor it out. Now you're going to do this no matter if it's a factor of this coefficient. So you might end up with a fraction here, but that's OK.
The point is is that we want to have 1x squared here. So I'm factoring out a 2. And notice that I've left some space here. And that's so I can construct my perfect square trinomial. That will eventually become a binomial squared.
So now I need to find out what that's going to be. So the next step is to take that x term coefficient. So in this case, it's 6. I'm going to divide it by 2 and then square it.
So 6 divided by 2 and then square it. That's going to give me 3 squared or 9. So what do I do with 9? Well, I'm going to use it to construct a perfect square trinomial. So I've added 9 here to this side of the equation.
Well, that means I need to add something here in order to keep this equal, but be careful, I cannot add just 0. That's not going to keep things equal, because there's this 2 outside of everything here. So really what they need to do is add 18 to the other side of the equation, because really I've added 18.
I've added 8 two times. So be careful about that. Let's go ahead and simplify then the left side of the equation. y minus 16 plus 18 is y plus 2.
And now because we have a perfect square trinomial, I can write this as a binomial squared. So this would be then 2 times x plus 3 quantity squared. Notice that if I double 3, I get the x term coefficient. And if I square root 3, I get the constant term here.
We're so close to vertex form the last step is to just express this as y equals. So we want y on one side. So I'm going to take away 2 from both sides of this equation. And I have y equals 2 times x plus 3 quantities squared minus 2.
And now this matches then our vertex form. We have y equals a times x minus h quantity squared plus k. We can see that our h value is negative 3 and our k value is negative 2.
So let's review converting a quadratic equation into vertex form. Well, we started with an equation in standard form. And we wanted to convert into vertex form. And we followed a process that involved completing the square.
So make sure that you have these steps written down in your notes. That's going to help you then convert from standard form into vertex form. Well, thanks for watching this tutorial. Hope to see you next time.