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

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Graphing Rational Functions", "Asymptotes of Rational Functions" ], "current_concepts": [ "Graphing Rational Functions", "Asymptotes of Rational Functions" ] } ## Identify the vertical asymptote $$ \begin{aligned} x + 4 &= 0 \ x &= -4 \end{aligned} $$ ## Identify the horizontal asymptote $$ \begin{aligned} \lim_{x \to \pm\infty} \frac{1}{x+4} &= 0 \ y &= 0 \end{aligned} $$ ## Match with the given graphs $$ \begin{aligned} &\text{Graph 1: Vertical asymptote at } x = 4 \ &\text{Graph 2: Vertical asymptote at } x = -4 \ &\text{Graph 3: Vertical asymptote at } x = 0 \text{ and shifted vertically} \ &\text{Therefore, the correct graph is the middle one (Graph 2).} \end{aligned} $$

(A) Left graph (vertical asymptote at $x = 4$)

(B) Middle graph (vertical asymptote at $x = -4$)

(C) Right graph (vertical asymptote at $x = 0$)

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Rational Functions" ] }

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Question

Explanation:

Identify the vertical asymptote

Identify the horizontal asymptote

Match with the given graphs

Answer: