Find the Domain y^2-4x+4y-4=0

Problem

y2−4*x+4*y−4=0

Solution

1. Identify the type of equation. This is the equation of a parabola that opens horizontally because the y term is squared and the x term is linear.

2. Isolate the variable x to express it as a function of y

y2+4*y−4=4*x

3. Divide both sides by 4 to solve for x

x=1/4*y2+y−1

4. Analyze the resulting function x=ƒ(y) Since this is a parabola opening to the right, the domain consists of all xvalues greater than or equal to the xcoordinate of the vertex.

5. Complete the square for the y terms in the original equation to find the vertex form.

y2+4*y+4=4*x+4+4

(y+2)2=4*x+8

(y+2)2=4*(x+2)

6. Determine the vertex (h,k) from the form (y−k)2=4*p*(x−h) Here, h=−2 and k=−2 so the vertex is (−2,−2)

7. Conclude the domain. Since the parabola opens to the right (4*p=4 which is positive), the xvalues start at the vertex x=−2 and extend to infinity.

Final Answer

Domain:[−2,∞)

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