In case of parabola

y= x^2-2x-3
Find:
a)Vertex:
b)Axis:
c)Focus:
d)Directrix:
e)Latus Rectum:

Y = x^2-2x-3.

a. h = Xv = -B/2A = 2/2 = 1.
K = Yv = 1^2-2*1-3 = -4.
V = (h,k) = (1,-4).

b. Axis: X = h = 1.

c. D(1,Y1), V(1,-4), F(1,Y2).

VF = Y2-(-4) = Y2+4 = 1/4a = 1/4.
Y2+4 = 1/4.
Y2 = 1/4-4 = 1/4-16/4 = -15/4.

d. DV = -4-Y1 = 1/4a = 1/4.
-4-Y1 = 1/4
-Y1 = 1/4+4 = 1/4+16/4 = 17/4.
Y1 = -17/4.

e. A(x1,-15/4), F(1,-15/4), B(x2,-15/4).

AF = 1-x1 = 1/2a = 1/2.
1-x1 = 1/2.
-x1 = 1/2-1 = -1/2
X1 = 1/2.

FB = x2-1 = 1/2a = 1/2.
X2 = 1/2+1 = 3/2.

NOTE: The Latus Rectum for the y-parabola is a hor. line that passes through the focus.

To find the vertex, axis, focus, directrix, and latus rectum of a parabola defined by the equation y = x^2 - 2x - 3, we can use the general form of the quadratic function:

y = ax^2 + bx + c,

where a, b, and c are constants.

a) Vertex:
The formula for finding the vertex of a parabola in the form of y = ax^2 + bx + c is given by:

Vertex x-coordinate = -b / (2a)
Vertex y-coordinate = f(x) = ax^2 + bx + c

In our equation, a = 1, b = -2, and c = -3. Plugging these values into the formula:

Vertex x-coordinate = -(-2) / (2 * 1) = 2 / 2 = 1
Vertex y-coordinate = f(1) = 1^2 - 2(1) - 3 = -4

Therefore, the vertex of the parabola is (1, -4).

b) Axis:
The axis of symmetry for a parabola is a vertical line passing through the vertex. Its equation is x = x-coordinate of the vertex.

Therefore, the axis of the parabola is x = 1.

c) Focus:
The formula for finding the focus of a parabola in the form of y = ax^2 + bx + c is given by:

Focus x-coordinate = -b / (2a)
Focus y-coordinate = f(x) + (1 - 4a) / (4a)

Plugging the values from the equation:

Focus x-coordinate = -(-2) / (2 * 1) = 2 / 2 = 1
Focus y-coordinate = -4 + (1 - 4(1)) / (4 * 1) = -3/4

Therefore, the focus of the parabola is (1, -3/4).

d) Directrix:
The equation of the directrix of a parabola in the form of y = ax^2 + bx + c is given by:

Directrix: y = f(x) - (1 + 4a) / (4a)

Plugging the values from the equation:

Directrix: y = -4 - (1 + 4(1)) / (4 * 1) = -17/4

Therefore, the directrix of the parabola is y = -17/4.

e) Latus Rectum:
The length of the latus rectum is given by |4a|, where a is the coefficient of the quadratic term.

In our equation, a = 1. So, the length of the latus rectum is |4(1)| = 4 units.

Therefore, the latus rectum of the parabola is 4 units.