![]() ![]() Now we've reduced the problem to a two-equations, two-unknowns one. Now set two pairs of equations equal to one another, eliminating C: equations 1 and 2, 2 and 3: If we solve all three equations for C, we get: We first use two sets of two equations to eliminate one of the variables ( C is usually easiest in these problems), then use those two equations to solve for the other two, and finally use the results to find the third variable. As per the rules of algebra, we must also add the same number to. Add (b/2) 2 to the quantity inside of the parenthesis. The coefficient in front of the first power term (x) is our value for b. We will convert to vertex form by completing the square. These three equations can be solved simultaneously by methods you already know. First, factor out the 9 from both x terms. Notice that none of these appears to be the vertex of the parabola, but that won't matter. It's probably best at this point to work an example or two, so let's find the equation of the parabola above, which passes through (-4, 8), (1, -4) and (3, -2). Now that's three equations and three unknowns ( A, B, C), so we should have enough information to determine what they are. If each of out three points satisfies this equation for a certain set of A, B and C (the things we're really looking for), we have three equations: ![]() Let's say that a parabola passes through three known points, $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3).$ Now we know that the general form of the equation we're looking for is The vertex (h, k) is located at h b 2a, k f(h) f( b 2a). The standard form of a quadratic function is f(x) a(x h)2 + k. The general form of a quadratic function is f(x) ax2 + bx + c where a, b, and c are real numbers and a 0. Here are the general forms of each of them: Standard form: f (x) ax 2 + bx + c, where a 0. The graph of a quadratic function is a parabola. We'd like to be able to find the equation of that parabola. A quadratic function can be in different forms: standard form, vertex form, and intercept form. It's a very specific kind of curve, the graph of a quadratic function. To convert a quadratic from y ax2 + bx + c form to vertex form, y a(x - h)2+ k, you use the process of completing the square. Now that we have values for a, h, and k, we can write a general equation for our parabola using vertex form Remember, vertex form is ya(x-h)2+k. That may be a bit hard to come to terms with at first, but remember that a parabola isn't just any curve. Through any three points, only one unique parabola can be drawn. ![]()
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