Find the Domain y^2-6y-10x+74=0

Problem

y^2−6*y−10*x+74=0

Solution

1. Identify the type of equation. This is a quadratic equation in terms of y and linear in terms of x representing a parabola that opens horizontally.

2. Isolate the term containing x to express x as a function of y

10*x=y^2−6*y+74

3. Solve for x by dividing both sides by 10

x=(y^2−6*y+74)/10

4. Analyze the range of the function x=ƒ(y) to determine the domain of the original relation. Since the coefficient of y^2 is positive, the parabola opens to the right, and the domain consists of all x values greater than or equal to the xcoordinate of the vertex.

5. Complete the square for the y terms to find the vertex.

x=(y^2−6*y+9−9+74)/10

x=((y−3)^2+65)/10

x=1/10*(y−3)^2+6.5

6. Determine the minimum value of x Since (y−3)^2≥0 the minimum value occurs at y=3 where x=6.5

7. State the domain based on the vertex (6.5,3) and the direction of opening.

Final Answer

Domain:[6.5,∞)

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