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what is the equation of the quadratic function that has a minimum at (7,-3) and goes through (9,9)?

f(x) = − +

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