Question 939490
Find the parabola that passes through points (3.5,10), (11,60), and (15,90)


{{{y=ax^2+bx+c}}} ...we need to find coefficients {{{a}}}, {{{b}}}, and {{{c}}}; so, use given points and set the system of three quations 

 for point (3.5,10) we have

{{{10=a(3.5)^2+b(3.5)+c}}}

{{{10=12.25a+(3.5)b+c}}}................eq.1


 for point(11,60)  we have

{{{60=a(11)^2+b(11)+c}}}

{{{60=121a+11b+c}}}................eq.2


for point (15,90) we have

{{{90=a(15)^2+b(15)+c}}}

{{{90=225a+15b+c}}}................eq.3


solve the system:

{{{10=12.25a+(3.5)b+c}}}................eq.1
{{{60=121a+11b+c}}}................eq.2
{{{90=225a+15b+c}}}................eq.3
------------------------------------------------------substitute eq.1 from eq.2

{{{60-10=121a+11b+c-(12.25a+(3.5)b+c)}}}................eq.2

{{{50=121a+11b+cross(c)-12.25a-(3.5)b-cross(c)}}}

{{{50=108.75a+7.5b}}} ...solve for {{{a}}}


{{{50-7.5b=108.75a}}}

{{{a=(50-7.5b)/108.75}}}.............1a


substitute eq.2 from eq.3


{{{90-60=225a+15b+c-(121a+11b+c)}}}................eq.3


{{{30=225a+15b+c-121a-11b-c}}}

{{{30=104a+4b}}}...solve for {{{a}}}


{{{a=(30-4b)/104}}}............2a

left sides in 1a and 2a are equal, so set  right sides   equal and solve for {{{b}}}

{{{(50-7.5b)/108.75=(30-4b)/104}}} ...cross multiply

{{{104(50-7.5b)=108.75(30-4b)}}}

{{{5200-780b=3262.5-435b}}}

{{{5200-3262.5=780b -435b}}}

{{{1937.5=345b}}}

{{{b=1937.5/345}}}

{{{b=5.615942028985507}}}

{{{b=5.6}}}

now find {{{a}}}

{{{a=(30-4b)/104}}}............2a

{{{a=(30-4*5.6)/104}}}

{{{a=(30-22.4)/104}}}

{{{a=7.6/104}}}

{{{a=76/1040}}}

{{{a=0.0730769230769231}}}

{{{a=0.07}}}


now we can find {{{c}}}

{{{10=12.25a+(3.5)b+c}}}................eq.1

{{{10=12.25*0.07+(3.5)5.6+c}}}

{{{10=0.8575+19.6+c}}}

{{{10=20.4575+c}}}

{{{10-20.4575=c}}}

{{{c=-10.4575}}}

{{{c=-10.5}}}

so, your equation is:

{{{highlight(y=0.07x^2+5.6x-10.5)}}}
(3.5,10), (11,60), and (15,90)

{{{drawing( 600, 600, -15, 100, -15, 100,
circle(3.5,10,.53),circle(11,60,.53),circle(15,90,.53), 
locate(3.5,10,p(3.5,10)),locate(11,60,p(11,60)),locate(15,90,p(15,90)),
graph( 600, 600, -15, 100, -15, 100, 0.07x^2+5.6x-10.5)) }}}