consider the following equation:
\f(x) = \frac{x^2 + 4}{4x^2 - 4x - 8}\

name the vertical asymptote(s).
$x = -1$ and $x = 2$
because
- $m

Question

consider the following equation:
\f(x) = \frac{x^2 + 4}{4x^2 - 4x - 8}\

name the vertical asymptote(s).
$x = -1$ and $x = 2$
because
- $m < n$
- $m = n$
- $a_m < b_n$
- $a_m = b_n$
- this is where the function is undefined

name the horizontal asymptote(s).
$y = 1/4$
because
- $m < n$
- $m = n$
- $a_m < b_n$
- $a_m = b_n$
- this is where the function is undefined

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Vertical Asymptotes", "Horizontal Asymptotes" ], "new_concepts": [], "current_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ] } ## Find vertical asymptotes and their justification $$ \begin{aligned} &f(x) = \frac{x^2+4}{4x^2-4x-8} = \frac{x^2+4}{4(x^2-x-2)} = \frac{x^2+4}{4(x-2)(x+1)}\ &\text{Denominator } 4(x-2)(x+1) = 0 \implies x = 2 \text{ or } x = -1\ &\text{Since } x^2+4 e 0 \text{ at these points, the function is undefined and has vertical asymptotes at } x = -1 \text{ and } x = 2. \end{aligned} $$ ## Find horizontal asymptote and its justification $$ \begin{aligned} &\text{Degree of numerator } (m) = 2\ &\text{Degree of denominator } (n) = 2\ &m = n \implies y = \frac{a_m}{b_n} = \frac{1}{4}\ &\text{The horizontal asymptote is } y = 1/4 \text{ because } m = n. \end{aligned} $$ ## Question 1 Name the vertical asymptote(s). x = -1 and x = 2 because $m < n$

$m = n$

$a_m < b_n$

$a_m = b_n$

this is where the function is undefined ## Question 2 Name the horizontal asymptote(s). y = 1/4 because $m < n$

$m = n$

$a_m < b_n$

$a_m = b_n$

this is where the function is undefined

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Horizontal Asymptotes" ] }

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QUESTION IMAGE

Question

Explanation:

Find vertical asymptotes and their justification

Find horizontal asymptote and its justification

Answer:

Question 1

Question 2