Question 1045127
Two points on the parabola are ( 5,  9/4 ) and ( 7, -15/4 ).
You do not yet know zeros or the actual vertex.  You DO know that {{{a<0}}}.


All you know about the vertex is some point  (h,3).  You can say the standard form equation is  {{{y=a(x-h)^2+3}}}.



<u>MAKE TWO SPECIFIC EQUATIONS USING THE "GIVEN" POINTS</u>


Still two variables are unknown, being h and a.  The two equations using the given points will allow to solve for h and a.


{{{system(a(5-h)^2+3=9/4,a(7-h)^2+3=-15/4)}}}


{{{system(a(5-h)^2=9/4-12/4,a(7-h)^2=-15/4-12/4)}}}


{{{system(a(5-h)^2=-3/4,a(7-h)^2=-27/4)}}}


Find the two expressions for "a".


{{{system(a=-3/(4(5-h)^2),a=-27/(4(7-h)^2))}}}


{{{4(5-h)^2/3=4(7-h)^2/27}}}


{{{27*4(5-h)^2/3=27*4(7-h)^2/27}}}


{{{36(5-h)^2=4(7-h)^2}}}


{{{9(5-h)^2=(7-h)^2}}}


{{{9(25-10h+h^2)=49-14h+h^2}}}


{{{225-90h+9h^2=49-14h+h^2}}}


{{{225-49-90h+14h+8h^2=0}}}


{{{176-76h+8h^2=0}}}


{{{highlight_green(4h^2-19h+44=0)}}}

{{{D=19^2-4*4*44=19^2-16*44=361-704}}}----------not working; unknown mistake.



I am leaving this incomplete and possibly incorrect solution posted anyway.  Maybe you understand the path taken and may do it properly and avoid whatever mistake I have made.



-
SHOULD be, the quadratic equation in just h,
{{{highlight_green(2h^2-19h+44=0)}}}
and then discriminant, {{{D=19^2-4*2*44=361-352=9=highlight_green(3^2)}}}, giving solution for h as
{{{h=(19+- 3)/(2*2)}}}
-
{{{h=(19+- 3)/4}}}
{{{system(h=4,or,h=11/2)}}}-----and maybe one of this might not work.


CORRESPONDING "a"?
Using either of the earlier equations solved for a,
{{{a=-3/(4(5-h)^2)}}}
-
{{{a=-3/(4(5-4)^2)}}}
{{{a=-3/(4*1)}}}
{{{a=-3/4}}}
-
{{{a=-3/(4(5-11/2)^2)}}}
{{{a=-3/(4(10/2-11/2)^2)}}}
{{{a=-3/(4(-1/2)^2)}}}
{{{a=-3/(4(1/4))}}}
{{{a=-3}}}
-
This gives two combinations of solutions of this system for a and h.
Either {{{system(a=-3/4,and,h=4)}}}
OR {{{system(a=-3,and,h=11/2)}}}.



WHAT IS THE STANDARD FORM EQUATION USING THESE?


Starting from {{{y=a(x-h)^2+3}}},


-------------------------------------------------------
{{{system(y=-(3/4)(x-4)^2+3,or,y=-3(x-11/2)^2+3)}}}
-------------------------------------------------------



--
The question asked was, what are the y-intercepts.  That not actually answered, but you could now easily just let y=0 and solve for x.  You may still need to do both equations separately.