Find the Equation of a Quadratic (Parabola) Given 3 Points

Say we're given these three points, 0, 11, 1, 5, and 2, 3. Now the equation of a parabola can be written in its general form like this, y equals ax squared plus bx plus c. So what we're going to do is we're going to solve for a, b, and c, okay? So what we need to do to solve for three variables is we need to have three equations. So let's see if we can write the three equations. So these are our x and y values. So when y is 11, right, x is 0. So that's going to be a times 0 squared plus b times 0 plus c, right? Okay, good. And you can see that anything times 0 is going to be 0. So we've already solved for c, so c is 11. Okay, now the next one, if y is 5, when x is 1, we have a times 1 squared plus b times 1 plus c. Or if you simplify this, this would be like a plus b plus c equals 5, okay? And then the last one, when y is 3, x is 2. So we're putting 2 in for x, so that's 2 squared plus b times 2 plus c. And if we simplify this down, we get 4 times a plus 2 times b plus c equals 3. I just flipped the equation to put the number on the right side. Okay, so you're with me so far? So we have three equations, and we're trying to solve for three variables. Now this one worked out kind of nice because we already know what c is, it's 11. So what I'm going to do is I'm just going to put the 11 back in here for c, and then what I'm going to do is I'm going to subtract it from both sides. So are you with me so far? So if I put 11 here, and I subtract 11, that's going to be negative 6. So we get a plus b equals negative 6. And over here, if I put in 11 and subtract, that's going to give us negative 8. So we have 4a plus 2b equals negative 8. So now we're down to two equations and two variables. So let's say we want to eliminate the b's. Let's multiply this entire equation by negative 2, okay, negative 2, and if we distribute, we're going to get negative 2a, okay, minus 2b equals positive 12, right, positive 12. So if we add these together, you get 2a, the b's cancel, and we get 4, and if we divide by 2, you can see that a equals 2. Okay, now how do we solve for b? Well, we can put a back in here, or here, or here, any one of these equations that has two variables. Let's put it in here. So if a is 2, we get 2 plus b equals negative 6, subtract 2, and you can see b is going to be negative 8. So now we have a, b, and c, we can put it back together into the equation here. So we've got y equals a, which is 2, so that's 2x squared, b, which is negative 8, so that's negative 8x, plus c, which is 11, and we have our equation in the general form, y equals ax squared plus bx plus c.