which of the following is the graph of this rational function?
\y = \frac{1}{x + 2}\

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Graphing Rational Functions", "Asymptotes of Rational Functions" ], "new_concepts": [], "current_concepts": [ "Graphing Rational Functions", "Asymptotes of Rational Functions" ] } ## Identify the asymptotes of the function Using the *Asymptotes of Rational Functions* knowledge point $$ \begin{aligned} &\text{Vertical asymptote: } x + 2 = 0 \implies x = -2\ &\text{Horizontal asymptote: } \lim_{x \to \pm\infty} \frac{1}{x+2} = 0 \implies y = 0 \end{aligned} $$ ## Analyze the behavior and key points Using the *Graphing Rational Functions* knowledge point $$ \begin{aligned} &\text{For } x > -2: y > 0 \quad (\text{e.g., } x = -1 \implies y = 1)\ &\text{For } x < -2: y < 0 \quad (\text{e.g., } x = -3 \implies y = -1) \end{aligned} $$ ## Match with the given options We examine the three graphs from left to right: - **First graph (left):** Has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 1$. This does not match. - **Second graph (middle):** Has a vertical asymptote at $x = -2$ and a horizontal asymptote at $y = 0$. For $x > -2$, the curve is in the upper half-plane ($y > 0$), and for $x < -2$, it is in the lower half-plane ($y < 0$). This matches our function perfectly. - **Third graph (right):** Has a vertical asymptote at $x = 3$ and a horizontal asymptote at $y = 0$. This does not match.

(A) Left graph: vertical asymptote at $x = 0$, horizontal asymptote at $y = 1$

(B) Middle graph: vertical asymptote at $x = -2$, horizontal asymptote at $y = 0$

(C) Right graph: vertical asymptote at $x = 3$, horizontal asymptote at $y = 0$

{ "elements": [ { "type": "line", "params": [[-2, -10], [-2, 10]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1.5, "dash": 2, "name": "x = -2", "withLabel": true, "labelPosition": [-1.8, 8] } }, { "type": "line", "params": [[-10, 0], [10, 0]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1.5, "dash": 2, "name": "y = 0", "withLabel": true, "labelPosition": [8, 0.3] } }, { "type": "functiongraph", "params": [{"js": "1/(x+2)", "latex": "\frac{1}{x+2}"}, -10, -2.05], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2.5 } }, { "type": "functiongraph", "params": [{"js": "1/(x+2)", "latex": "\frac{1}{x+2}"}, -1.95, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2.5 } }, { "type": "point", "params": [[-1, 1]], "properties": { "name": "(-1, 1)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[-3, -1]], "properties": { "name": "(-3, -1)", "size": 3, "color": "#5583F2", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Rational Functions" ] }

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Question

Explanation:

Identify the asymptotes of the function

Analyze the behavior and key points

Match with the given options

Answer: