which of the following is the graph of $(x - 3)^2 + (y + 1)^2 = 9$?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Circle Equation", "Graphing Circles" ], "new_concepts": [], "current_concepts": [ "Circle Equation", "Graphing Circles" ] } ## Identify the center and radius from the equation $$ \begin{aligned} &(x - h)^2 + (y - k)^2 = r^2 \ &(x - 3)^2 + (y + 1)^2 = 9 \ &h = 3, \quad k = -1 \implies \text{Center: } (3, -1) \ &r^2 = 9 \implies r = 3 \end{aligned} $$ ## Determine the key boundary points of the circle $$ \begin{aligned} &\text{Right: } (3 + 3, -1) = (6, -1) \ &\text{Left: } (3 - 3, -1) = (0, -1) \ &\text{Top: } (3, -1 + 3) = (3, 2) \ &\text{Bottom: } (3, -1 - 3) = (3, -4) \end{aligned} $$ ## Match with the correct graph option - Left graph: Center is at $(3, -1)$, passes through $(0, -1)$ on the y-axis, and goes down to $y = -4$. - Middle graph: Center is at $(3, 1)$, which is incorrect. - Right graph: Center is in the second quadrant, which is incorrect.

(A) The first graph (left) with center at $(3, -1)$ and radius $3$

(B) The second graph (middle) with center at $(3, 1)$

(C) The third graph (right) with center at $(-3, 1)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Graphing Circles" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the center and radius from the equation

Determine the key boundary points of the circle

Match with the correct graph option

Answer: