The line y=3x-3 is tangent to a parabola which has vertex (0,0) and axis of symmetry the y-axis.   Find the equation of the parabola.

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kjcdb8er | Teacher | (Level 1) Associate Educator

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The general equation of a parabola in Cartesian coordinates is y = ax^2 + bx + c. Since the vertex (h,k) is at (0,0), we know that c = 0  ( plug in x = 0 and y = 0 ).

we also know that since h = -b/4a, b = 0

We also know that somewhere on this parabola, the line y = 3x - 3 is tangent to the parabola. Tangent means that the slopes are the same. Let the line touch the parabola at the point (s,t). So we know the following:

t = 3s - 3

t = a(s^2)

3 = 2a*s     ( because the slope of the parabola is y' = 2ax + b )

Here you have 3 equations and 3 unknowns, so solve:

t = 3s/2 = 3s - 3  --> 3s = 6s - 6 --> s = 2

thus, t = 3 and a = 3/4

So, y = 3/4 x^2

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