The function f(t) = t2 + 4t − 14 represents a parabola. Part A: Rewrite the function in vertex form by completing the square. Show your work. (6 points) Part B: Determine the vertex and indicate whether it is a maximum or a minimum on the graph. How do you know? (2 points) Part C: Determine the axis of symmetry for f(t). (2 points)

A)  The vertex ( h , k) = ( -2 , -18)

B) The minimum value of the given function = - 18

C) The Axis of the symmetry for f(t) is y -axis

Step-by-step explanation:

A)

Given a parabola   f(t) = t² + 4 t − 14

                               f(t) =  t² + 2(2) t + (2)²-4− 14

                           f(t) = (t +2)² - 18

Let comparing  y = (x +2)² -18

                   (x +2)² = y + 18

                   (x-h))² = 4 a ( y - k))²

The vertex ( h , k) = ( -2 , -18)

B)

  Given a parabola   f(t) = t² + 4 t − 14

  Differentiating with respective to 't'

                                 f¹(t) = 2 t + 4

                                  f¹(t) = 2 t + 4 = 0

now                      

Again Differentiating with respective to 't'

f(x) has a minimum value at t = -2

Given f(t) = t² + 4 t − 14

         f( -2) = 4 + 4(-2) -14 = 4 -8 -14 = -18

The minimum value of the given function = - 18

C)

f(t) = (t +2)² - 18

Let comparing  y = (x +2)² -18

                   (x +2)² = y + 18

                   (x-h))² = 4 a ( y - k))²

The vertex ( h , k) = ( -2 , -18)

The Axis of the symmetry for f(t) is y -axis