Question

The function g(n) = n2 − 20n + 95 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

`n^2 - 20n + 95=\\ n^2-20n+100-5=\\ (n-10)^2-5`

Vertex -
`(10,-5)`
It's a minimum, because
`a\ \textgreater \ 0`

The axis of symetry is x=h where h is the x-coordinate of the vertex.
So it's
`x=10`

Answer 2

`g(n)=(n-10)^2-5`

The vertex will be at
`(10, -5)`

`x=10`

Explanation:

The given quadratic function is,

`g(n) = n^2-20n+95`

`=n^2-2\cdot n\cdot 10+95`

`=(n^2-2\cdot n\cdot 10+10^2)-10^2+95`

`=(n-10)^2-100+95`

`=(n-10)^2-5`

The vertex form is,

`g(n)=(n-10)^2-5`

The vertex will be at
`(10, -5)`

As the leading coefficient of
`g(n) = n^2-20n+95` is positive, so the parabola will open upwards. Hence, at the vertex the value will be minimum.

The axis of symmetry will be,

`x=-(b)/(2a)`

Putting the values,

`x=-(-20)/(2* 1)=10`