Question 1190797

 axis parallel to Ox=>your parabola opens right/left

Vertex form: 

{{{(y-k)^2=4p(x-h)}}}

vertex on O{{{y}}}=> {{{h=0}}}

{{{(y-k)^2=4p(x-0)}}}

{{{(y-k)^2=4px}}}............passing through ({{{2}}}, {{{2}}}) , use it to calculate {{{k}}} and {{{p}}}

{{{(2-k)^2=4p*2}}}

{{{k^2 - 4 k + 4=8p}}}

{{{p= (k^2 - 4 k + 4)/8}}}..........1)


passing through ({{{8}}}, {{{-1}}})


{{{(-1-k)^2=4p*8}}}

{{{k^2 + 2 k + 1=32p}}}

{{{p=(k^2 + 2 k + 1)/32}}}..........2)


from 1) and 2) we have


 {{{(k^2 - 4 k + 4)/8=(k^2 + 2 k + 1)/32}}}

 {{{32(k^2 - 4 k + 4)=8(k^2 + 2 k + 1)}}}

{{{32 k^2 - 128 k + 128-8k^2 - 16 k -8=0}}}

{{{24 k^2 - 144 k + 120=0}}}

{{{24 (k^2 - 6k + 5)=0}}}

{{{24 ((k - 1) (k - 5))=0}}}


=> {{{k=1 }}}or {{{k=5}}}


then, if {{{k=1}}}

{{{p= (1^2 - 4*1 + 4)/8}}}..........1)

{{{p= 1/8}}}

or, if {{{k=5}}}

{{{p= (5^2 - 4*5 + 4)/8}}}

{{{p=9/8}}}


and your equations are:

{{{(y-1)^2=4(1/8)x}}}

{{{(y-1)^2=(1/2)x}}}.........1

or

{{{(y-5)^2=4(9/8)x}}}

{{{(y-5)^2=(9/2)x}}}.........2