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Question 1118375: Find the equation of the parabola that concaves down in the form of y = a(x-h)^2 + k, given that the vertex is (-3,35) and a point on the parabola is (-5,23).
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
You are given the form for the answer: 
In that form, the vertex of the parabola is (h,k). Since you are given that the vertex is (-3,35), the equation is of the form
The usual way of finding the value of the coefficient a is to plug in the x and y values of the other known point on the parabola:
Then the complete equation is
If you understand a bit about parabolas and quadratic functions, you can find the value of a without plugging in the x and y values of the other point. Here is the shortcut; you can see where the shortcut comes from by looking at the detailed steps used above to find the value of a.
(1) The given point that is not the vertex is 2 units away in the x direction from the vertex (right or left doesn't matter because of the symmetry of the parabola).
(2) In the equation, the x term is squared; so square the distance from (1): 2^2 = 4.
(3) The y value, from the vertex to the other point, changed by -12, from 35 to 23.
(4) The coefficient a is -12/4 = 3.
This is similar to what you know about slopes of linear equations.
In the linear equation y = ax+b, the coefficient a is the slope; it is the change in y divided by the change in x.
For a quadratic equation, in which the x term is SQUARED, the coefficient a is the change in y divided by the SQUARE of the change in x.
To go through the calculation again for your example, we have:
change in x from vertex to other point: 2
change in y from vertex to other point: -12
coefficient a: -12/(2^2) = -12/4 = -3
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