Part A:
[tex]h(x)= x^{2} +14x+41[/tex]
The first step of completing the square is writing the expression [tex] x^{2} +14x[/tex] as [tex] (x+7)^{2} [/tex] which expands to [tex] x^{2} +14x+49[/tex].
We have the first two terms exactly the same with the function we start with: [tex] x^{2} [/tex] and [tex]14x[/tex] 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 [tex] x^{2} +14x+49[/tex]
The function in completing the square form is
[tex]h(x)= (x+7)^{2}-8 [/tex]
Part B:
The vertex is obtained by equating the expression in the bracket from part A to zero
[tex]x+7=0[/tex]
[tex]x=-7[/tex]
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 [tex]a x^{2} +bx+c[/tex] can either make U-shape or ∩-shape depends on the value of the constant [tex]a[/tex] that goes with [tex] x^{2} [/tex]. When [tex]a[/tex] is (+), the curve is U-shape. When [tex]a[/tex] (-), the curve is ∩-shape
Part C:
The symmetry line of the curve will pass through the vertex, hence the symmetry line is [tex]x=-7[/tex]
This function is shown in the diagram below
