Writing quadratic functions given 3 points

0:00- [Instructor] What I want to do is find the equation for a parabola that passes through these three points. Or another way you could think about it, the equation of a quadratic that passes through these three points. So, pause this video. Have a go at this before we do this together.

0:17All right, so the key here is to recognize that an, the general equation for quadratic, or if you think in the graph terms for parabola, is going to be some constant A times x squared plus some other constant B times X plus some other constant C. And what we could do is use each of these points to set up a new equation with three unknowns.

0:41The unknowns are going to be A, B, and C. And we're going to have three equations with three unknowns, which we can then solve multiple ways. I'll solve it with substitution. So, let's look at this first XY pair. So let's use that to set up our first equation.

0:58When Y is equal to negative five, we have A times negative one squared. Negative one squared is one. So that's just going to be A. And then negative one times B, so A minus B, and then we have plus C. All right, now let's do our second equation here. X is one, Y is negative one.

1:19So when Y is negative one, I'm just substituting in here, we have A times one squared, so that's just going to be equal to A, plus B times one. So that's just plus B. And then we have plus C. And then last but not least, we have, let me pick a nice color here.

1:38This salmon color seems nice. When X is two, Y is four. So another way is Y is four. So four is equal to A times two squared, so that's four A and then plus B times two, so plus two B, and then we have plus C at the end, plus C. So there you have it, a system of three equations and three unknowns.

2:04How do we solve this? We could solve it with substitution, we can solve it with elimination. I will tackle elimination. And the way that you could do it is you can use one of the equations to eliminate a variable with each of the other ones. So for example, I could start with, let me.

2:23If I were to add both sides of these two equations right over here, that would eliminate the B variable. When I add these two sides, these two are going to cancel out. So actually let me do that. So if I add the left hand sides, I'm going to get negative six and then is equal to the right hand side, A plus A is two A.

2:44Negative B and B cancel out. And then I have plus two C. So the next thing I want to do is then, let me, well, I'll just use a neutral white, is then I can try to also eliminate B using these two equations. Now there it's not quite as easy. I'm gonna have to multiply this yellow equation.

3:05Let's see, if I multiply it by negative two, that will work out. So once again, this is the same equation, but if I multiply it by negative two, so times negative two, I'm going to get positive two is equal to negative two A minus two B minus two C. And so I'm going to use these two.

3:23I can now add both sides. If you add the left hand side, you get six is equal to four A plus four A minus two A is going to be equal to two A. And then C minus two C is equal to minus C. So now I have two equations with two unknowns. And now what variable do I wanna eliminate next?

3:52Well, let's see. I could, if I multiply both sides of this white equation by a negative one, then the two As we can eliminate. So let's do that. Multiply both sides by negative one, so this becomes a negative six. This becomes a negative two A, this becomes A plus C.

4:07And now if I add, I'm running outta colors. If I add the sides to each other, negative six plus negative six is negative 12 is equal to, these cancel, at two C plus C is equal to three C. Divide both sides by three, and I get C is equal to negative four. So, that is good.

4:29We have figured out what C is equal to, negative four. Let me not write it that way. I'll write it later. C is equal to negative four. And now we can substitute back into, let's say, this equation. Well substitute back into this equation right over here to figure out what A is going to be.

4:46So, we're going to have, let me write it in purple here. Negative six is equal to two A and then plus two times negative four, times negative four. And so another way of saying that, that means that, and I'm running outta space a little bit. Negative six is equal to two A minus eight.

5:11Let me add eight to both sides. And I get two A is equal to two. Did I do that right? Yeah. And then divide both sides by two. And I get A is equal to one, making progress, A is equal to one. And then we could go back to maybe that, any equation, but we could go to that top equation and then solve for B.

5:37We have negative five is, that's a different blue. Let me see. Negative five, negative five is equal to A, which is one, minus B plus C, C is negative four. So we're adding a negative four there. And so we have negative five is equal to, we could write this as, well let's say negative three minus B, I just added the one and the negative four.

6:02I could multiply every time, everything times negative one if I like. So I get five is equal to, I could write B plus three, and you could subtract three from both sides to get that B is equal to two. So there you have it. We know what A, B, and C is. We can now write our equation.

6:21We know that the quadratic, the equation of the parabola or the quadratic that goes through these three points is Y is equal to A-X squared, A is one. So let me write this down. It's one, one X squared plus B is two plus two times X plus C, which is negative four.

6:49So instead of plus negative four, I am just going to write a, I'm just going to over here, write a minus four. And we're done. There you have it. That's our equation for our parabola or for our quadratic.