select the correct answer.

the graphs of rational functions $m$ and $n$ have the same vertical asymptotes, and both have a single $x$-intercept at $x = 5$.
which equation could represent function $n$?

$n(x) = 5m(x)$
$n(x) = m(x) + 5$
$n(x) = m(x - 5)$
$n(x) = m(x + 5)$

Question

select the correct answer.

the graphs of rational functions $m$ and $n$ have the same vertical asymptotes, and both have a single $x$-intercept at $x = 5$.
which equation could represent function $n$?

$n(x) = 5m(x)$
$n(x) = m(x) + 5$
$n(x) = m(x - 5)$
$n(x) = m(x + 5)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Functions", "Vertical Asymptotes", "Function Transformations" ], "new_concepts": [], "current_concepts": [ "Rational Functions", "Vertical Asymptotes", "Function Transformations" ] } ## Analyze the properties of the rational functions $$ \begin{aligned} &\text{Let } m(x) = \frac{p(x)}{q(x)} \text{ where } p(5) = 0 \text{ and } q(x) = 0 \text{ defines the vertical asymptotes.}\ &\text{Since } n(x) \text{ has the same vertical asymptotes, its denominator must share the same roots as } q(x).\ &\text{Since } n(x) \text{ has the same single } x\text{-intercept at } x = 5\text{, its numerator must share the root } x = 5. \end{aligned} $$ ## Evaluate the given transformation options $$ \begin{aligned} &1.\ n(x) = 5m(x) \implies \text{Vertical stretch by a factor of 5. Denominator and roots remain unchanged.}\ &2.\ n(x) = m(x) + 5 \implies \text{Vertical shift. Changes the } x\text{-intercepts.}\ &3.\ n(x) = m(x - 5) \implies \text{Horizontal shift right by 5. Shifts asymptotes and intercepts.}\ &4.\ n(x) = m(x + 5) \implies \text{Horizontal shift left by 5. Shifts asymptotes and intercepts.} \end{aligned} $$ ## Determine the correct representation $$ \begin{aligned} &n(x) = 5m(x) \text{ preserves both the vertical asymptotes and the } x\text{-intercept at } x = 5. \end{aligned} $$

(A) $n(x) = 5m(x)$

(B) $n(x) = m(x) + 5$

(C) $n(x) = m(x - 5)$

(D) $n(x) = m(x + 5)$

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

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QUESTION IMAGE

Question

Explanation:

Analyze the properties of the rational functions

Evaluate the given transformation options

Determine the correct representation

Answer: