Determine the quadratic function f whose graph is given The

Determine the quadratic function f whose graph is given. The vertex is (3, -2) and the other given point is (1,6).

Solution

f(x) = ax² +bx +c ==> f \'(x) = 2ax + b

At vertex (3 , -2) function has minimum value

==> f \'(x) at (3 , -2) is zero

==> f \'(3) = 0

==> 2a(3) + b = 0

==> b = -6a --------- (1)

(1 , 6) ==> 6 = a(1)² +b(1) +c

==> a + b + c = 6

from (1) a + (-6a) + c = 6

==> -5a + c = 6 --------------- (2)

(3 , -2) ==> -2 = a(3)² + b(3) + c

==> 9a +3b + c = -2

from (1) ==> 9a +3(-6a) + c = -2

==> -9a + c = -2 --------- (3)

Solving (2) and (3)

(2) - (3) ==> -5a + c - (-9a + c) = 6 - (-2)

==> -5a + c + 9a - c = 8

==> 4a = 8 ==> a = 2

from (2)   -5(2) + c = 6

==> c = 16

from (1) b = -6(2) = -12

Hence f(x) = 2x² - 12x + 16 is required equation.