In case of parabola

y= x²-2x-3

Find:

a)Vertex:

b)Axis:

c)Focus:

d)Directrix:

e)Latus Rectum:

In case of parabola

y= x²-2x-3

Now,

y= x²-2x-3

x²-2x+1=y-3-1

(x-1)²=y-4......(i)

Let x-1= X and y+4=Y

Thus, the equation of parabola becomes, X²=Y......(ii)

Equating equation (ii) with the standard equation, X²=4aY

⇒4a = 1

⇒a = 1/4

Focus

X=0  and  Y = 1/4

x-1= 0  and  y+1=1/4

x=1  and  y=(-3/4)

FOCUS= (1,-3/4)

Vertex 

X=0  and  Y = 0

x-1=0  and  y+4=0

x=1  and  y = -4

VERTEX=(1,-4)

Axis:  

Since the given equation involves y , the axis of the parabola is the x-axis.

Latus Rectum

=4a 

=4×(1/4)

=1

LATUS RECTUM = 1

Directrix

y+4+(1/4)=0

4y+17=0

DIRECTRIX=4y+17=0