I am not sure what the answer would be for this problem?! Not sure which one to pick! :c PLEASE HELP

Question

Which equation correctly solves for h, the x-coordinate of the vertex of a quadratic function, when starting with the equation for the vertex form of a parabola?

A) h=x+√y-k/a
B) h=x+√a-y/k
C) h=x+√y+k/a
D) h=x+√a-k/y

Answer 1

      The vertex form for the equation of a parabola is y =a(x - h)² + k  where the point (h,k) is the vertex of the  parabola.;  i.e. the lowest point (or highest point) located a x = h and y = k.

 

       Doing a little algebra one can shift the equation around to (x - h)² = ( y - k)/a    (Note:  A word of caution here.  The right hand side of the equation could be negative if you pick a value of y that is outside of its range.  Don't worry about this just yet, we well address it later.)

 

      Now,  take the square root of both sides  ( x - h) = ± √[(y - k)/a]   (Note: Every item on the rhs is under the √.)  

 

and solve for h:   h = x ± √[(y - k)/a]       This is the best, and correct,  solution for  h.  (But be careful.)

 

 

Comment:  The original equation was  y = a(x -h)² + k .  Depending on the sign of coefficient a, the parabola either opens upwards a›0, or downwards, a<0.  The trivial case of a = 0 is just a horizontal line, which has no vertex.   The original equation has a domain of all real numbers.  Its range is more limited, determined by the values  of a and k.  One must take care before blindly plugging values for y into our solution for h.  These values must match the original equation; basically (y - k)/a must be ≥ 0.

 

 

Comment: When one takes the square root of an equation  one runs the risk of introducing extraneous or "mirror-image" solutions.  By convention we usually choose the +√ of  ±√ . This usually works the best.

Answer 2

Fred:

 

I think sqrt y - k/a in the question should be sqrt[(y - a)/k],  otherwise no given answer is correct.

 

Equation to a parabola in vertex format is  y = a(x - h)² + k.

 

A) gives sqrt(y - k/a) = h - x  = - (x - h)   (Note: Write h-x as -(h-x)  trick question?).

 

 

then, (y - k)/a = (x - h)^2,

 

and y = a(x - h)² + k.

 

Bingo!  The correct answer is (A)

 

Dr. G.