Question

The function g(n) = n2 − 6n + 16 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 g(n). (2 points)

Answer 1

Part A

After dividing the first two terms by the coefficient of n², the coefficient of the linear term is -6, so we can complete the square by adding (and subtracting) the square of half that: (-6/2)² = 9.

... g(n) = n² -6n + 9 + 16 - 9

... g(n) = (n -3)² +7 . . . . . . . rewrite to vertex form

Part B

The generic vertex form is

... y = a(x -h)² +k . . . . . . for vertex (h, k) and vertical expansion factor "a"

Comparing this to g(n), we see a=1, h=3, k=7. When a > 0, the parabola opens upward, and the vertex is a minimum. Here, we have a > 0, so we can conclude ...

... the vertex (3, 7) is a minimum

Part C

The axis of symmetry is the vertical line through the vertex.

... x = 3