Question

How do you find the vertex of the parabola: `y=0.5x^2+2x`?

Answer 1

The vertex is at `(-2,-2)`

Explanation:

The general vertex form of a quadratic is
`color(white)("XXXX")` `y =m(x-a)^2+b`
`color(white)("XXXX")` `color(white)("XXXX")`with its vertex at `(a,b)`

To convert `y=0.5x^2+2x` into vertex form:

Extract `m`
`color(white)("XXXX")` `y = (0.5)(x^2+4x)`

Complete the square
`color(white)("XXXX")` `y = (0.5)(x+4x+4) - (0.5)(4)`

Re-write as a squared binomial and simplify the constant
`color(white)("XXXX")` `y = (0.5)(x+2)^2 - 2`

Write in explicit vertex form
`color(white)("XXXX")` `y = (0.5)(x-(-2))^2 + (-2)`
`color(white)("XXXX")` `color(white)("XXXX")`with vertex at `(-2,-2)`