Question

What is the vertex of `y= -2x^2+8x-(x-1)^2`?

Answer 1

Vertex at `(x-v,y_v)=(1 2/3, 7 1/3)`

Explanation:

Convert the given equation `y=-2x^2+8x-(x-1)^2`
into vertex form:
`color(white)("XXX")y=color(green)m(x-color(red)a)^2+color(blue)b` with vertex at `(color(red)a,color(blue)b)`

`y=-2x^2+8x-(x-1)^2`

`color(white)("XXX")=-2x^2+8x-x^2+2x-1`

`color(white)("XXX")=-3x^2+10x-1`

`color(white)("XXX")=color(green)(-3)(x^2-10/3x)-1`

`color(white)("XXX")=color(green)(-3)(x^2-10/3x+((cancel(10)^5)/(cancel(6)_3))^2)-1-(color(green)(-3)) * (5/3)^2`

`color(white)("XXX")=color(green)(-3)(x-color(red)(5/3))^2-1+25/3`

`color(white)("XXX")=color(green)(-3)(x-color(red)(5/3))^2+color(blue)(22/3)`

which is the vertex form with the vertex at
`color(white)("XXX")(color(red)(5/3),color(blue)(22/3))=(color(red)(1 2/3),color(blue)(7 1/3))`