The function h(x) = x2 + 14x + 41 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 h(x). (2 points)

Part A:



The first step of completing the square is writing the expression  as  which expands to .

We have the first two terms exactly the same with the function we start with:  and  but we need to add/subtract from the last term, 49, to obtain 41. 

So the second step is to subtract -8 from the expression

The function in completing the square form is


Part B:

The vertex is obtained by equating the expression in the bracket from part A to zero




It means the curve has a turning point at x = -7

This vertex is a minimum since the function will make a U-shape. 
A quadratic function  can either make U-shape or ∩-shape depends on the value of the constant  that goes with . When  is (+), the curve is U-shape. When  (-), the curve is ∩-shape

Part C:

The symmetry line of the curve will pass through the vertex, hence the symmetry line is 

This function is shown in the diagram below

Iagree that is the answer to the question