What is the equation of this parabola and the value of z?

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SUMMARY

The equation of the parabola can be determined using the points A (17, 3), B (51, 3), and C (34, z), where C is a local minimum. By applying the derivative condition y' = 2ax + b and substituting the coordinates of the points, three simultaneous equations can be established to solve for the coefficients a, b, and c. The symmetry of the points indicates that the vertex is at (34, z), allowing for the calculation of z using either of the other two points.

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Hey guys! I'm new here & browse more thanb I post
I have an emergency, and a lot of guys who work in IT are pretty knowledgeable.
If you could help me out with my question below I would be so happy! Thanks in advance,
Damien.

Question:

I need the EQUATION of a PARABOLA in the form of y = ax^2 + bx + c
I have three points; A (17, 3), B (51,3) and C (34, z)
Z is unkown, however, point C is a local minimum.
So y' = 2ax + b

I need the EQUATION of a PARABOLA in the form of y = ax^2 + bx + c
I have three points; A (17, 3), B (51,3) and C (34, z)
Z is unkown, however, point C is a local minimum.
So y' = 2ax + b. Can anyone help me find the equation of the parabola and the value of z?
 
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You appear to have written the question twice.

Just write out the equations after substituting the values of the points, along with the y' as you have written and solve them for a,b,c simultaneously. To make things a little easier, you can use row-reduction (linear algebra) to solve for it easily.
 
Since C(34,z) is a local minimum, you must have f '(34)= 0. That gives you one equation in a and b. Putting (17,3) and (51,3) for (x,y) give you two more. You can solve the 3 equations for a, b, and c. That's one way to do the problem.

Notice, however, the symmetry: Since (17,3) and (51,3) have the same y value, they must be equal distances from the vertex: (17+ 51)/2= 68/2= 34. You can, of course, write a parabola as y= (x- a)2+ c where (a, c) is the vertex (here (34, z). Since you know a= 34, use either of the other two points to solve for c= z.
 

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