Question

Question #7c401

Answer 1

The vertex is at `"(-1,2)"`.

Explanation:

The vertex of a parabola is the midpoint between the focus and the point where the axis (the axis = the line perpendicular to the directrix, and pasding through the focus), intersects the directrix.

1) Focus = `color(blue)("(-2,1)")`

2) Axis = perpendicular to the directrix through the focus: the directrix has slope `-1` so the perpendicular line has slope `+1` (product of slopes of perpendicular lines = `-1`). The line with slope `+1` through `color(blue)("(-2,1)")` is `color(red)(y=x+3)`.

3) Intersection of the axis with the dierctrix:

`color(red)(y=x+3)`
`x+y=3`

Thus the intersection is at `color(green)("(0,3)")`.

4) Midpoint between the focus `color(blue)("(-2,1)")` and `color(green)("(0,3)")` is

`"(-1,2)"`.