Question

What is the axis of symmetry and vertex for the graph `y = x^2 – 4x – 12`?

Answer 1

Axis of symmetry `-> x= +2`

`"Vertex" ->(x,y)=(2,-16)`

Explanation:

`color(blue)("Using a bit of a cheat to find "x_("vertex"))`

Given`" "y=x^2color(magenta)(-4)x-12`.....................Equation (1)

`ul("Axis of symmetry is the x value of the vertex")`

`color(green)(x_("vertex")=(-1/2)xx(color(magenta)(-4)) = +2)`

'.....................................................................................................
`color(brown)("A note about what I have just done:")`
Consider the standard form `y=ax^2+bx+c`

Write as `y=a(x^2+b/ax)+c`

Then `x_("vertex")=(-1/2)xxb/a`

In the case of this question `a=1`
'..................................................................................................

`color(blue)("Determine "y_("vertex"))`

Substitute `x=2` into Equation (1)

`color(brown)(y=x^2-4x-12" "->" "y_("vertex")=(color(blue)(2))^2-4(color(blue)(2))-12`

`color(green)(y_("vertex")=4-8-12 = -16)`

`"VERTEX" ->(x,y)=(2,-16)`