draw the lines representing the vertical and horizontal asymptotes on the graph of this function.

(f(x) = \frac{2x^2 + 7x - 4}{x^2 + 5x + 4})

Question

draw the lines representing the vertical and horizontal asymptotes on the graph of this function.

(f(x) = \frac{2x^2 + 7x - 4}{x^2 + 5x + 4})

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ], "current_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ] } ## Simplify the rational function $$ f(x) = \frac{2x^2 + 7x - 4}{x^2 + 5x + 4} = \frac{(2x - 1)(x + 4)}{(x + 1)(x + 4)} = \frac{2x - 1}{x + 1}, \quad x eq -4 $$ ## Determine the vertical asymptote $$ \begin{aligned} &x + 1 = 0 \ &x = -1 \end{aligned} $$ ## Determine the horizontal asymptote $$ \begin{aligned} &y = \lim_{x \to \pm\infty} \frac{2x^2 + 7x - 4}{x^2 + 5x + 4} = \frac{2}{1} \ &y = 2 \end{aligned} $$ The equations of the asymptotes are: Vertical asymptote: $x = -1$ Horizontal asymptote: $y = 2$ { "elements": [ { "type": "line", "params": [[-1, -10], [-1, 10]], "properties": { "name": "x = -1", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "line", "params": [[-10, 2], [10, 2]], "properties": { "name": "y = 2", "strokeColor": "#8C55F2", "strokeWidth": 2, "dash": 2, "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Simplify the rational function

Determine the vertical asymptote

Determine the horizontal asymptote

Answer: