finding a polynomial function

(3,2) (8, 7) and (1, -3)

We can generate a quadratic polynomial for this, thusly:

a(3)^2 + b(3) + c  = 2  →   9a + 3b + c = 2   (1)

a(8)^2 + b(8) + c  = 7 →  64a + 8b + c = 7   (2)

a(1)^2 + b(1) + c = -3 →      a + b + c = -3  (3)

Subtract (1) from (2) and (3) from (2).....this give us

55a + 5b  = 5   →  11a + b = 1   multiply through by -7  →  -77a - 7b  = -7   ( 4) 

63a + 7b = 10  →   63a + 7b  = 10   (5)

Add (4)  and (5)

-14a  = 3 

a  = -3/14

And using 11a + b = 1 to find b, we have

11(-3/14) + b = 1

-33/14 + b = 1

b = 47/14

And using (3) to find c, we have

-3/14 + 47/14 + c = -3

44/14 + c = -3

22/7 + c = -3

c = -43/7

So....the polynomial is

P(x)  =  (-3/14)x^2 + (47/14)x - 43/7