Find the Properties (x-2)^2=-8(y+3)

Problem

(x−2)2=−8*(y+3)

Solution

1. Identify the form of the parabola. The equation is in the standard form (x−h)2=4*p*(y−k) which represents a parabola that opens vertically.

2. Determine the vertex (h,k) by extracting the values from the equation.

(x−2)2=−8*(y+3)

h=2

k=−3

Vertex=(2,−3)

3. Calculate the value of p by setting the coefficient of the linear term equal to 4*p

4*p=−8

p=−2

4. Determine the direction of opening. Since the equation starts with (x−h)2 and p<0 the parabola opens downward.

5. Find the focus using the formula (h,k+p)

Focus=(2,−3+(−2))

Focus=(2,−5)

6. Find the directrix using the formula y=k−p

y=−3−(−2)

y=−1

7. Identify the axis of symmetry using the formula x=h

x=2

8. Calculate the length of the latus rectum using |4*p|

Length=|−8|

Length=8

Final Answer

Vertex: *(2,−3), Focus: *(2,−5), Directrix: *y=−1, Axis of Symmetry: *x=2, Direction: Downward

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