Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y=x^2-4x-5`?

Answer 1

Axis of Symmetry: `x=2`
Minimum value: `y=-9` at point `(2, -9)`

Explanation:

First, factor the equation to find the roots:
`y=x^2-4x-5` Set `y=0` to find the roots of the equation.
`0=x^2-4x-5` Factor
`(x+1)(x-5)=0` Using the zero products property,
`(x+1)=0` and `(x-5)=0` so the roots are:
`x=-1, 5`

Since `a` of `ax^2+bx+c` for this equation is positive, it opens upward with a minimum value, which is below `0` because it has `2` roots. Since parabolas are symmetric, the axis of symmetry must be in the middle of the two roots:
A.o.S.`=((x_1+x_2)/2)` (adapted average formula)
A.o.S.`=(((-1)+5)/2)`
A.o.S.`=2`
Axis of Symmetry: `x=2`

The axis of symmetry will intersect the minimum of the parabola, so we can input `x=2` into the equation:
`y=x^2-4x-5`
`y=(2)^2-4(2)-5` Combining like terms:
`y=4-8-5` Combining like terms:
`y=-9`
Therefore, the minimum is `(2, -9)`