what are the equations of the asymptotes of the graph of the function $f(x) = \frac{3x^2 - 2x - 1}{x^2 + 3x - 10}$?

$x = -5$, $x = 2$ and $y = 3$
$x = -2$, $x = 5$ and $y = 3$
$x = 3$, $y = -5$, and $y = 2$
$x = 3$, $y = -2$, and $y = 5$

Question

what are the equations of the asymptotes of the graph of the function $f(x) = \frac{3x^2 - 2x - 1}{x^2 + 3x - 10}$?

$x = -5$, $x = 2$ and $y = 3$
$x = -2$, $x = 5$ and $y = 3$
$x = 3$, $y = -5$, and $y = 2$
$x = 3$, $y = -2$, and $y = 5$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vertical Asymptotes", "Horizontal Asymptotes" ], "new_concepts": [], "current_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ] } ## Find the vertical asymptotes Using the *Vertical Asymptotes* knowledge point $$ \begin{aligned} &x^2 + 3x - 10 = 0 \ &(x + 5)(x - 2) = 0 \ &x = -5 \quad \text{or} \quad x = 2 \end{aligned} $$ Since the numerator $3x^2 - 2x - 1$ is non-zero at these values: $$ \begin{aligned} &3(-5)^2 - 2(-5) - 1 = 75 + 10 - 1 = 84 eq 0 \ &3(2)^2 - 2(2) - 1 = 12 - 4 - 1 = 7 eq 0 \end{aligned} $$ The vertical asymptotes are $x = -5$ and $x = 2$. ## Find the horizontal asymptote Using the *Horizontal Asymptotes* knowledge point $$ \begin{aligned} &\lim_{x \to \pm\infty} \frac{3x^2 - 2x - 1}{x^2 + 3x - 10} = \lim_{x \to \pm\infty} \frac{3 - \frac{2}{x} - \frac{1}{x^2}}{1 + \frac{3}{x} - \frac{10}{x^2}} = 3 \end{aligned} $$ The horizontal asymptote is $y = 3$.

(A) $x = -5$, $x = 2$ and $y = 3$

(B) $x = -2$, $x = 5$ and $y = 3$

(C) $x = 3$, $y = -5$, and $y = 2$

(D) $x = 3$, $y = -2$, and $y = 5$

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

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Find the vertical asymptotes

Find the horizontal asymptote

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