Analyze x²-6x-5y=1. Determine the vertex, focus, directrix, intercepts, axis of symmetry, and at least 2 other points on the graph.

Answer 1

X^2 - 6X - 5Y = 1.

Solve for Y:

-5Y = -X^2 + 6X +1,
Divide both sides by -5:
Eq2: Y = X^2/5 - 6X/5 + 1/5,

h = Xv = -b/2a = (6/5) / (2/5) = 3.

Substitute 3 for X in Eq2:
k = Yv = 3^2/5 - 6*3/5 + 1/5,
k = 9/5 - 18/5 + 1/5,
k = 9/5 - 17/5 = - 8/5 = -1 3/5.

V(h , k) = V(3 , - 1 3/5 )

Axis = h = 3.

F(3 , Y2)

V(3 , - 1 3/5)

D(3 , Y1)

4a = 4(1/5) = 4/5.

1/4a = 5/4 = 1 1/4.

Y2 = K + 1/4a,
Y2 = - 8/5 + 5/4,
Common denominator = 20:
Y2 = - 32/20 + 25/20 = - 7/20.


Y1 = K - 1/4a,
Y1 = - 8/5 - 5/4,
Y1 = -8/5 - 5/4,
Common denominator = 20:
Y1 = -32/20 - 25/20,
Y1 = -57/20 = -2 17/20.

F(3 , -7/20)

V(3 , -1 3/5)

D(3 , -2 17/20)

P1(2 , -1 2/5)

P2(4 , -1 2/5)

X-Intercepts = 5.8 , 0.17.
Use Quadratic Formula.

Answer 2

CORRECTION: The constant in Eq2 should
be NEGATIVE 1/5.

Y = X^2/5 - 6X/5 - 1/5,

h = Xv = -b/2a = (6/5) / (2/5) = 3.

Substitute 3 for x in Eq2:
k=Yv=3^2/5 - 6*3/5 - 1/5 = - 10/5 =-2.

V(h , k) = V(3 , -2).

Axis = h = 3.

4a = 4(1/5 = 4/5).

1/4a = 5/4 = 1 1/4.

F(3 , Y2)

V(3 , -2)

D(3 , Y1)

Y2 = K + 1/4a = -2 + 5/4 = -3/4.

Y1=K-1/4a = -2 - 5/4 = -13/4 = -3 1/4.

F(3 , -3/4)

V(3 , -2)

D(3 , -3 3/4)

P1(2 , -1 4/5)

P2(4 , -1 4/5)

X-Intercepts = 6.16 and -0.16.