which of the following is the graph of this rational function?
\y = \frac{1}{x} + 4\

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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 rational function Using the *Asymptotes of Rational Functions* knowledge point $$ \begin{aligned} &\text{Vertical asymptote: } x = 0\ &\text{Horizontal asymptote: } y = 4 \end{aligned} $$ ## Analyze the behavior of the branches Using the *Graphing Rational Functions* knowledge point $$ \begin{aligned} &\text{For } x > 0: y = \frac{1}{x} + 4 > 4\ &\text{For } x < 0: y = \frac{1}{x} + 4 < 4 \end{aligned} $$ ## Match with the given graph options Using the *Graphing Rational Functions* knowledge point $$ \begin{aligned} &\text{First graph: Vertical asymptote at } x = 0\text{, Horizontal asymptote at } y = 4.\ &\text{Second graph: Vertical asymptote at } x = 5\text{, Horizontal asymptote at } y = 0.\ &\text{Third graph: Vertical asymptote at } x = 0\text{, Horizontal asymptote at } y = -4. \end{aligned} $$

(A) The first graph (vertical asymptote at $x = 0$, horizontal asymptote at $y = 4$)

(B) The second graph (vertical asymptote at $x = 5$, horizontal asymptote at $y = 0$)

(C) The third graph (vertical asymptote at $x = 0$, horizontal asymptote at $y = -4$)

{ "elements": [ { "type": "line", "params": [[-10, 4], [10, 4]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1.5, "dash": 2, "name": "y = 4", "withLabel": true, "labelPosition": [8, 4.5] } }, { "type": "line", "params": [[0, -10], [0, 10]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1.5, "dash": 2, "name": "x = 0", "withLabel": true, "labelPosition": [0.5, 8] } }, { "type": "functiongraph", "params": [{"js": "1/x + 4", "latex": "\frac{1}{x} + 4"}, 0.1, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2.5 } }, { "type": "functiongraph", "params": [{"js": "1/x + 4", "latex": "\frac{1}{x} + 4"}, -10, -0.1], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2.5 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Rational Functions" ] }

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Question

Explanation:

Identify the asymptotes of the rational function

Analyze the behavior of the branches

Match with the given graph options

Answer: