Question

What is the axis of symmetry and vertex for the graph `y=-x^2+x+12`?

Answer 1

`color(blue)( "Axis of symmetry "-> x=1/2)`

`color(green)("Vertex "->" "(x,y)" "->" " (1/2,12 1/4)`

Explanation:

It is not uncommon for people to be shown the method of completing the square to solve this context. At first it is quite confusing so I am going to show you something that is part way towards completing the square as an alternative.

Given:`" "y=-x^2+x+12`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Compare to `y=ax^2+bx+c`

Rewritten as:`" "a(x^2+b/ax)+c`

Then you have:`" "x_("vertex")=(-1/2)xxb/a`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In your case
`a=(-1)`
`b=(+1)`

So we have:

`color(blue)(x_("vertex") = (-1/2)xx1/(-1) = +1/2)`
`color(blue)( "Axis of symmetry "-> x=1/2)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute `x=1/2` in the original equation and you should end up with:

`color(blue)(y_("vertex")= 12 1/4)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("Vertex "->" "(x,y)" "->" " (1/2,12 1/4)`