# How do you find the vertex, focus and directrix of # x^2-4x+y+3=0#?

##### 1 Answer

Vertex

Focus

Directrix

#### Explanation:

Complete the square.

#y = -x^2 + 4x -3#

#= -(x -2)^2 + 1#

The vertex is at

The parabola is concave down, as the leading coefficient

#a = -1 < 0#

graph{-x^2+4x-3 [-10, 10, -5, 5]}

Therefore, the focus lies

#p = frac{1}{4abs(-1)} = 1/4#

The

The coordinates of the focus is

All points on the parabola are equidistant from the focus and the perpendicular to the directrix.

We know that the directrix is a horizontal line, with the equation

We consider the distance of the vertex to the focus, it is

#m = 1 + 1/4 = 5/4#

The equation of the directrix is