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

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Question 939490: Find the parabola that passes through points (3.5,10), (11,60), and (15,90)
Answer by MathLover1(20849) About Me  (Show Source):
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Find the parabola that passes through points (3.5,10), (11,60), and (15,90)

y=ax%5E2%2Bbx%2Bc ...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%283.5%29%5E2%2Bb%283.5%29%2Bc
10=12.25a%2B%283.5%29b%2Bc................eq.1

for point(11,60) we have
60=a%2811%29%5E2%2Bb%2811%29%2Bc
60=121a%2B11b%2Bc................eq.2

for point (15,90) we have
90=a%2815%29%5E2%2Bb%2815%29%2Bc
90=225a%2B15b%2Bc................eq.3

solve the system:
10=12.25a%2B%283.5%29b%2Bc................eq.1
60=121a%2B11b%2Bc................eq.2
90=225a%2B15b%2Bc................eq.3
------------------------------------------------------substitute eq.1 from eq.2
60-10=121a%2B11b%2Bc-%2812.25a%2B%283.5%29b%2Bc%29................eq.2
50=121a%2B11b%2Bcross%28c%29-12.25a-%283.5%29b-cross%28c%29
50=108.75a%2B7.5b ...solve for a

50-7.5b=108.75a
a=%2850-7.5b%29%2F108.75.............1a

substitute eq.2 from eq.3

90-60=225a%2B15b%2Bc-%28121a%2B11b%2Bc%29................eq.3

30=225a%2B15b%2Bc-121a-11b-c
30=104a%2B4b...solve for a

a=%2830-4b%29%2F104............2a
left sides in 1a and 2a are equal, so set right sides equal and solve for b
%2850-7.5b%29%2F108.75=%2830-4b%29%2F104 ...cross multiply
104%2850-7.5b%29=108.75%2830-4b%29
5200-780b=3262.5-435b
5200-3262.5=780b+-435b
1937.5=345b
b=1937.5%2F345
b=5.615942028985507
b=5.6
now find a
a=%2830-4b%29%2F104............2a
a=%2830-4%2A5.6%29%2F104
a=%2830-22.4%29%2F104
a=7.6%2F104
a=76%2F1040
a=0.0730769230769231
a=0.07

now we can find c
10=12.25a%2B%283.5%29b%2Bc................eq.1
10=12.25%2A0.07%2B%283.5%295.6%2Bc
10=0.8575%2B19.6%2Bc
10=20.4575%2Bc
10-20.4575=c
c=-10.4575
c=-10.5
so, your equation is:
highlight%28y=0.07x%5E2%2B5.6x-10.5%29
(3.5,10), (11,60), and (15,90)