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Anonymous asked in Science & MathematicsMathematics · 1 month ago

# Is it possible to determine the equation of a quadratic function given the vertex and one other point on the parabola? ?

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certainly...equation is "  y - b = K ( x - a)² " where ( a , b) is the vertex...so only K is unknown and the one point will determine that

• Only if you assume the parabola is in standard position and not rotated.

y = ax² + bx + c

You have 3 unknowns and only two points on the locus of the curve. That is not sufficient to solve for the three unknowns.

If you assume that the parabola is in standard position, you can use that assumption as the third condition required to make the problem determinate.

Ans:

In the general case, knowing only the vertex and one point on the locus is NOT sufficient to determine the equation of the parabola. You must have one more known condition to make the problem determinate. That third condition may be the unstated assumption that the rotation angle of the parabola is zero. But understand that is an assumption and is not necessarily true in the general case.

• Yes, if the vertex is identified AS the vertex.

• Let's say you put the equation in the form  y = ax^2 + bx + c

You know two points (x0, y0) and (x1, y1)  where (x0, y0) are the vertex.  So

y0 = ax0^2 + bx0 + c

y1 = ax1^2 + bx1 +c

But there are three unknowns a, b, and c.  We need one more equation.  We can use the fact that the slope of a line tangent to the vertex is zero or from calculus:

dy/dx = 0 = 2ax + b at x = x0  -->  2ax0 = -b  so now

y0 = a*x0^2 - 2ax0*xo + c

y1 = a*x1^2 - 2ax0*x1 + c

and you can solve these for a and c.  So it is possible

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