Answer:
[tex] P(x) = x^2 + 4x + 4[/tex]
Step-by-step explanation:
For this case the quadratic function is given by this formula:
[tex] P(x) = ax^2 + bx + c[/tex]
On this case we have the following points (0,4) ,(1,9) , (-3,1) so then we have the following 3 equations:
[tex] 4 = a(0)^2 +b(0) + c = c[/tex] (1)
From the equation (1) we have the value for c on this case c=4
[tex] 9 = a(1)^2 +b(1) + 4= a +b+4 [/tex] (2)
[tex] 1 = a(-3)^2 +b(-3) + 4= 9a-3b +4 [/tex] (3)
We can rewrite equations (2) and (3) like this:
[tex] 5= a+b [/tex] (2)
[tex] -3 = 9a-3b[/tex] (3)
If we find a from equation (2) we got:
[tex] a = 5-b[/tex] and we replace this into equation (3) we got:
[tex] -3 = 9(5-b) -3b= 45-9b -3b = 45-12b[/tex]
[tex] 12b = 48, b = 4[/tex]
And then [tex] a = 5-4 = 1[/tex]
And then our equation for the polynomial would be:
[tex] P(x) = x^2 + 4x + 4[/tex]
