I really need help to set this one up. Not sure where to start

Question

Use a system of equations to find the quadratic function  f(x) = ax2 + bx + c
that satisfies the equations. Solve the system using matrices.
f(1) = 5, f(2) = 16, f(3) = 31

 

f(x) =

Answer 1

Begin with f(x) = ax^2 + bx + c.

 

 

5 = a(1)^2 + b(1) + c    implies 5 = a + b + c            Equation (1)

 

16 = a(2)^2 + b(2) + c    implies 16 = 4a + 2b + c     Equation (2)

 

31 = a(3)^2 + b(3) + c    implies 31 = 9a + 3b + c     Equation (3)

 

This is 3 equations in 3 variables. We will solve for a,b, and c.

 

We can eliminate c by subtracting (1) from (2) to obtain:  3a + b = 11        Equation (4)

 

We can eliminate c by subtracting (1) from (3) to obtain:  5a + b = 15        Equation (5)

 

If we subtract (4) from (5) we get 2a = 4. Therefore, a = 2.

 

By substitution with 2 in place of a in (5) gives us 5(2) + b = 15 which means 10 + b = 15 which means that b = 5.

 

Finally, using equation (1) with a = 2, and b = 5 yields that c = -2.

 

Thus, f(x) = 2x^2 + 5x - 2.