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