Question 184560: I'm can't figure out how to create a quadratic equation out of the vertex and a point that it must pass through. All i've managed so far is some trial and error and come up with this.
y=2(x+2)²+4
From the vertex of (-2,4) and passing through (-1,8).
I have been fiddling around with it and have two ideas how to proceed.
I could change (x+2) to something else to move it along the x axis or change the first part of it to change the fatness or whichever you want to call it of the parabola. Regardless i figure there is a better way to do it other then trial and error, if it has been answered before i'm sorry I didn't see it. I'm self taught and this is the first time I really have had any problem.
-Thanks
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
If you have three points on a parabola, you can determine the equation of the parabola by substituting the coordinates of your points into:
Giving you three simultaneous linear equations that can be solved for the values of the coefficients A, B, and C.
But you only have two points, namely the vertex and one other point. However, if you use the properties of symmetry for a parabola, you can easily develop the coordinates of the third point.
Your only problem is that you must decide whether your parabola's axis of symmetry is vertical, giving you an equation where y is defined in terms of x, or whether the axis is horizontal, giving you an equation where x is defined in terms of y.
If (-2, 4) is the vertex and (-1,8) is a point on the parabola with a vertical axis of symmetry, there is another point on the parabola with a y value of 8 and the x value that produces it will be equidistant from the vertex on the other side of the vertex from (-1,8), namely (-3, 8).
If this is the case, then:
)
)
)
Solve this system by your favorite method; either Gaussian Elimination or Cramer's Rule should do. The solution set will give you the three required coefficients.
If you think your parabola has a horizontal axis, then your third point is (-1, 0), and you will use:
to set up your three equations.
In the absence of any evidence as to the orientation of the axis of symmetry, you might consider doing the problem both ways.
John
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