Question

What is the axis of symmetry and vertex for the graph `y = -x^2 - x + 9`?

Answer 1

Axis of symmetry: x=-0.5
Vertex: (-0.5,9.75)

Explanation:

Factorising to find roots:
`-(x^2+x-9)` (I took out the -1 because I find it easier to factorise without that extra negative in there confusing things)
`-(x+5)(x-4)`
`x=-5, x=4`

Half way between these points is the axis of symmetry and the vertex.
Total distance between the points: 9
Half that: 4.5
So the axis of symmetry is at `x=(-5+4.5)= -0.5`

Now we also know the x value of the vertex: -0.5. Substituting this back into the original equation will give the y value:
`-(-0.5)^2-(-0.5)+9=y`
`0.5^2+0.5+9=y`
`0.25+0.5+9=y`
`y=9.75`

Therefore vertex at `(-1/2, 9.75)`

graph{-x^2-x+9 [-7, 7, -15, 10]}