Answer: The equation is y = 3*x^2 - 42*x + 144
Step-by-step explanation:
A quadratic function has the shape:
a*x^2 + b*x + c
and the minimum needs a to be positive and the minumum can be obtained by:
2ax + b = 0
x = -b/(2a)
and we know that the minimum is at x = 7
7 = -b(2a)
from this we get:
b = -7*2a, now, we can replace it in the previous equation, and remember that when x = 7, we have y = -3
a*7^2 + (-7*2a)*7 + c = -3
a*49 - a*98 + c = -3
and when x = 9, we have that y = 9
a*9^2 + (-7*2a)*9 + c = 9
a*81 - a*126 + c = 9
so we have two equations and two variables, those are:
a*49 - a*98 + c = -3
a*81 - a*126 + c = 9
first, we can isolate c in one of the equtions and replace it on the other.
c = 9 -a*81 + a*126 = 9 + a*(126 - 81) = 9 + a*45
now we replace it in the other equation:
a*49 - a*98 + 9 + a*45 = -3
a*(49 - 98 + 45) + 9 = - 3
a*(-4) + 9 = -3
a*(-4) = -12
a = 12/4 = 3
then we can replace it in:
c = 9 + a*45 = 9 + 3*45 = 144
and b = -7*2a = -7*2*3 = -42
then la equation is:
y = 3*x^2 - 42*x + 144
