Final answer:
The correct answer for the translated graph equation is (x-2)(y-3)=6, representing the graph of xy=6 translated up by 3 units and to the right by 2 units. The correct answer is option C.
Step-by-step explanation:
The question revolves around the translation of a hyperbola defined by the equation xy=6. When a graph is translated, each point on the graph moves in a specified direction by a certain amount. To translate the graph up by 3 units and to the right by 2 units, we would adjust the x and y coordinates of each point on the original graph accordingly. To reflect this transformation mathematically, we modify the original equation to compensate for the shifts: the x-coordinate is reduced by 2, and the y-coordinate is increased by 3.
The correctly translated equation that represents this transformation is (x-2)(y-3)=6. This is because for every point (x, y) on the original graph, the corresponding point on the translated graph will be (x+2, y+3). Therefore, plugging in x-2 in place of x and y-3 for y will yield points that satisfy the original equation.
If we address each possible answer:
- Option A: y = 3 + 6/(x-2) is incorrect because it represents a vertical shift but not a horizontal translation of the hyperbola.
- Option B: y = 3x/(x-2) is incorrect as it misrepresents the transformation and the original relationship between x and y.
- Option C: (x-2)(y-3) = 6 is the correct representation of the graph translated up by 3 units and to the right by 2 units.
- Option D: y = 3/(x-2) only reflects a horizontal translation, not the vertical shift needed.