Write an quadratic equation in vertex form
And then what's the y-coordinate of the vertex?
Standard form of quadratic equation
And y is equal to They are in different forms. So, where do we hit a maximum point? This one in particular is going to be an upward opening parabola, and so it might look something like this. So let's say, I'm just gonna make this up, we have y is equal to negative pi times x minus 2. And so to find the y value of the vertex, we just substitute back into the equation. Well, we hit a maximum point when x minus five is equal to zero, when we're not taking anything away from the I have to be very careful here.
The whole point of this is that now I can write this in an interesting way. And we talk about where that comes from in multiple videos, where the vertex of a parabola or the x-coordinate of the vertex of the parabola.
This whole thing is going to hit a minimum value when this term is equal to 0 or when x equals 2. So the whole point of this is now to be able to inspect this.
I could have literally, up here, said hey, I'm adding 20 and I'm subtracting And so, x minus five is equal to zero. So your minimum point for this curve right over here, for your parabola, is going to happen when this expression is equal to zero, when you're not adding anything to negative And I know its graph is going to be a parabola.
This coordinate right over here is the point 2, negative 5. So it's five comma 10 and our curve is gonna look something like this.
Standard form to vertex form worksheet
As you might remember from other videos, if we have a quadratic, if we're graphing y is equal to some quadratic expression in terms of x, the graph of that will be a parabola, and it might be an upward opening parabola or a downward opening parabola. And a is the coefficient on the x squared term. Well here, this is gonna be downward opening and let's appreciate why that is. So now, you hopefully appreciate why this is called vertex form. So let me rewrite that. And so the vertex here is x equals five, and I'm just gonna eyeball it, maybe it's right over here, x equals five. The important thing to realize is that this part of the expression is never going to be negative. It's quite straightforward to pick out the vertex when you have something written in this way. I don't know exactly where it intersects the x-axis but it's going to be a downward opening parabola. Well, if x is equal to five and this thing is zero, you're not gonna be taking anything away from the 10 and so y is going to be equal to Well, this is going to be equal to positive 20 over 10, which is equal to 2. And we'll see where this comes from when you look at the quadratic formula. I could write this as y is equal to 5 times x minus 2 squared, and then 15 minus 20 is minus 5. Well, the x-coordinate is going to be the x value that makes this equal to zero, which is 2. This one in particular is going to be an upward opening parabola, and so it might look something like this.
Let me just rewrite it again. So it is 5 times x squared minus 4x.
I don't know actually where this does intersect the x-axis or if it does it all. Now it's not so satisfying just to plug and chug a formula like this. So let me rewrite that. And a is the coefficient on the x squared term. If you were to distribute this, you'll see that. Now, there's many ways to find a vertex. So let's say, I'm just gonna make this up, we have y is equal to negative pi times x minus 2. The y value is going to be 5 times 2 squared minus 20 times 2 plus 15, which is equal to let's see. Probably the easiest, there's a formula for it. So if I want to turn something that looks like this, 2ax, into a perfect square, I just have to take half of this coefficient and square it and add it right over here in order to make it look like that. Because the coefficient on the x squared term here is positive, I know it's going to be an upward opening parabola. So now, you hopefully appreciate why this is called vertex form.
I have to be very careful here.
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