Two parabolas have the same x- intercepts (-2,0) & (4,0). The max/min values of the first parabola is two times the max/min value of the other parabola. Determine the equations of the two parabolas.
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The general equation of a parabola is y = ax^2 + bx + c
The question gives the x-intercepts of the parabola as (-2,0) and (4,0).
0 = a(x - 4)(x + 2)
=> a(x^2 - 2x - 8) = 0
=> ax^2 - 2ax - 8a = 0
A parabola either has a minimum point or a maximum point. It cannot have both. If the parabola opens upwards it has a minimum point and if it opens downwards it has a maximum point.
Let the minimum of one parabola be twice that of the other. Then we have two equations: y = a1*x^2 - 2a1*x - 8*a1 = 0 and y = a2*x^2 - 2a2*x - 8*a2 = 0.
Any equation of a parabola where a1 = 2*a2 will satisfy the condition.
An example of two such parabolas is provided below.
There can be an infinite number of parabolas that satisfy the given condition. They have a general equation y = ax^2 - 2ax - 8a where the ratio between a for the two equations is 2.
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