Question

The function h(x) = x2 + 14x + 41 represents a parabola.

Answer 1

to anyone who may be a lil' sus, the previous answer is correct.

Answer 2

Part A:

$h(x)= x^{2} +14x+41$

The first step of completing the square is writing the expression $x^{2} +14x$ as $(x+7)^{2}$ which expands to $x^{2} +14x+49$.

We have the first two terms exactly the same with the function we start with: $x^{2}$ and $14x$ 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 $x^{2} +14x+49$

The function in completing the square form is
$h(x)= (x+7)^{2}-8$

Part B:

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

$x+7=0$
$x=-7$

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 $a x^{2} +bx+c$ can either make U-shape or ∩-shape depends on the value of the constant $a$ that goes with $x^{2}$. When $a$ is (+), the curve is U-shape. When $a$ (-), the curve is ∩-shape

Part C:

The symmetry line of the curve will pass through the vertex, hence the symmetry line is $x=-7$

This function is shown in the diagram below