Question 558739
The equation is the equation of a parabola in vertex form.
VERTEX
The coordinates of the vertex are shown in the equation subtracted from the {{{x}}} and the {{{y}}}.
The vertex is (2,-3).
The vertex has {{{x-2=0}}} <---> {{{x=2}}} and {{{y+3=0}}} <---> {{{y=-3}}}, and {{{x-2=0}}}.
It is a minimum, because, for any value of {{{x}}} other than {{{x=2}}}, {{{(x-2)^2>0}}}, making {{{y+3>0}}} <---> {{{y>-3}}}.
AXIS OF SYMMETRY
For all points other than the vertex, the same value of {{{y}}} happens for two different values of {{{x}}}, at equal distances to the left and right of the line {{{x-2=0}}} <---> {{{x=2}}}. That line is the axis of symmetry.
FOCUS AND DIRECTRIX
The focus is the point (2,-3+c) above the vertex/minimum that the parabola "wraps" around. The directrix is the line {{{y=-3-c}}} at the same distance on the other side of the vertex.
Your book will tell you that the coefficient of {{{y}}} in the equation equals {{{4c}}}, so in this case {{{1=4c}}} --> {{{c=1/4}}}
The focus has {{{y=-3+c=-3+1/4=-11/4}}}. It is the point (2,-11/4).
The directrix is the line {{{y=-3-c=-3-1/4}}} --> {{{y=-13/4}}}.