How Do You Correctly Derive the Equation of a Parabola with a Horizontal Axis?

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SUMMARY

The correct derivation of the equation of a parabola with a horizontal axis involves shifting the origin to the vertex point (h, k) and applying the transformation correctly. The equation in the new coordinate system is y² = 4ax, which translates to (y + k)² = 4a(x + h) in the original coordinate system. The key error in the initial derivation was the assumption that merely shifting the origin suffices without considering the need for a coordinate rotation when transforming from a vertical to a horizontal parabola.

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Hi! I am trying to find the equation of a parabola with vertex as (h,k) and axis parallel to the x-axis. However, I am not able to derive the correct result.
(1) I shift the origin to the point (h,k).
(2) Now the equation of the parabola in the new system becomes y^{2} = 4ax.
(3) Now, we know that when we shift the origin without rotation of axes to a point (h,k) (wrt the old system), the locus in the old system is changed by replacing x by x-h and y by y-k. This gives us the equation of the locus in the new system.
Now, since the equation of the parabola in the new system is y^{2}=4ax, then the equation of the parabola in the old system must be (y+k)^{2} = 4a(x+h), as on replacing x by x-h and y by y-k in this equation, we get the equation of the parabola in the new system, which is y^{2} = 4ax.
Please help in finding the fault in this derivation. Thanks!
 
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Do you want to transform a parabola whose axis is along the y-axis ( f(x) = 4a x^2 ) to one whose axis is parallel to the new x-axis? You will have to rotate the coordinate system, not simply move the origin. What form do you want the new parabola to take?
 

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