Question

How many points does `y=-2x^2+x-3` have in common with the vertex and where is the vertex in relation to the x axis?

Answer 1

`x_("vertex")=+1/4`

The number of points in common with the vertex and the graph is 1

Explanation:

This is a quadratic equation and the coefficient of `x^2` is negative. Consequently the graph is of form `nn` thus the vertex is a maximum

`color(red)("If")` the coefficient has been positive then you would have the general form of `uu`

The vertex has only one point.

`color(green)("A sort of cheat to determine x-vertex")`
Not really a cheat as it is part of the process for completing the square.

Write as: `color(darkcyan)(-2)(x^2+color(magenta)(1/(-2)) x)-3`

Note that `(color(darkcyan)(-2))xx(color(magenta)(-1/2))x=+x`

`x_("vertex")=(-1/2)xxcolor(magenta)(-1/2) = +1/4`