To find the vertex form of the quadratic function h(x) = x² + 14x + 6, we need to complete the square.
First, we can start by rewriting the function:
h(x) = x² + 14x + 6
Next, we focus on the quadratic and linear terms, which are x² + 14x. To complete the square, we take the coefficient of x, which is 14, divide it by 2 to get 7, and then square it to get 49.
Now, we rewrite the function by adding and subtracting this square:
h(x) = (x² + 14x + 49) – 49 + 6
Now, we can simplify this:
h(x) = (x + 7)² – 49 + 6
h(x) = (x + 7)² – 43
Thus, the vertex form of the quadratic function is h(x) = (x + 7)² – 43.
This tells us that the vertex of the parabola represented by this function is at the point (-7, -43).