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The points at which the parabola intersects the x axis are 3-sqrt(10)/2 and 3+sqrt(10)/2. The X position of the vertex is in the middle, at 3. The y position, from there, is simply found by substituting 2 for x in the equation. As a result, the vertex is at (3, 5).
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the equation of a parabola is: y = a(x-h)^2 + k *h and k are the x and y intercepts of the vertex respectively * x and y are the coordinates of a known point the curve passes though * solve for a, then plug that a value back into the equation of the parabola with out the coordinates of the known point so the equation of the curve with the vertex at (0,3) passing through the point (9,0) would be.. 0 = a (9-0)^2 + 3 = 0 = a (81) + 3 = -3/81 = a so the equation for the curve would be y = -(3/81)x^2 + 3
So you need something like this: y = a*(x - 4)² + 3. This will make the vertex be at (4,3). Then it looks like you have another point on the parabola (3,5). Plug that in and solve for a. 5 = a*(3-4)² + 3. This becomes 5 = a + 3, so a=2, then the equation is: y = 2*(x - 4)² + 3