Directrix and focus of a x 2 + b x + cHow can you find the directrix and focus of a parabola (quadratic function) a x 2 + b x + c ,where a ≠ 0 ? I mean, given the focus x,y and directrix (I'll use a horizontal line for simplicity) y = k you can find the equation of the quadratic; how do you do this backwards?

Question

Directrix and focus of   a      x    2    +  b  x  +  cHow can you find the directrix and focus of a parabola (quadratic function)  a      x    2    +  b  x  +  c  ,where   a  ≠  0  ? I mean, given the focus x,y and directrix (I&#039;ll use a horizontal line for simplicity)   y  =  k you can find the equation of the quadratic; how do you do this backwards?

kavdawg8w8 · Accepted Answer

Step 1Okay, so for to answer your question I started off with making a standard parabola from the given equation.  y  =  a      x    2    +  b  x  +  c  y  −  c  +            b      2              4              a        2              =  a      x    2    +  2  (            b      a              2      a        )  x  +            b      2              4              a        2              (  a  x  +      b          2      a            )    2    =  y  −  c  +            b      2              4              a        2              (  x  +      b          2              a        2                  )    2    =      1          a      2        (  y  −  c  +            b      2              4              a        2              )This gives you a standard equation in form of  (  x  −  H      )    2    =  4  A  (  y  −  B  )Thus it&#039;s clear that your Vertex of this parabola are :  (            −      b              2              a        2              ,  c  +            −              b        2                    4              a        2              )Step 2Now, Let an equation be      X    2    =  4  Ywhere  X  =  x  −  Hand  Y  =  y  −  Bfor this parabola,   A  =  1 Hence it&#039;s focus is  (  0  ,  −  1  )implying  X  =  0and  Y  =  −  1to get the actual focus, simply substitute X and Y respectivel according to values given to get actual focus on  (  H  ,  B  −  1  )where H and B are already specified. Now to get your directrix,put  Y  −  1  =  0to get actual directrix as  y  =  B  +  1

Celeste Barajas · Accepted Answer

Step 1The goal is essentially to get   y  =  a      x    2    +  b  x  +  c into the form   4  p  (  y  −  k  )  =  (  x  −  h      )    2  . This can be done by completing the square:                    y                            =        a                  x          2                +        b        x        +        c                                                  y            −            c                    a                                    =                  x          2                +                  b          a                x                                                  y            −            c                    a                +                              (                          b                              2                a                                      )                    2                                    =                  x          2                +                  b          a                x        +                              (                          b                              2                a                                      )                    2                                              1          a                          (          y          −          c          +                                    b              2                                      4              a                                )                                    =                              (            x            +                          b                              2                a                                      )                    2                .            Step 2Thus the parabola can be written as  4  ⋅      1          4      a            (    y    −          (      c      −                        b          2                          4          a                    )        )    =            (      x      −              (        −                  b                      2            a                          )            )        2    .The focus is      (    −          b              2        a              ,    c    −                  b        2                    4        a              +          1              4        a              )    =      (    −          b              2        a              ,                  1        −                  b          2                            4        a              +    c    )  and the directrix is  y  =  c  −            b      2              4      a        −      1          4      a        =  c  −            1      +              b        2                    4      a        .

Directrix and focus of ax^2+bx+c

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