which function has a horizontal asymptote of $y = 3$?

$f(x) = 3(2^x)$
$f(x) = 2(4)^{x-3}$
$f(x) = 2(3^x)$
$f(x) = 2(4^x) + 3$

Question

which function has a horizontal asymptote of $y = 3$?

$f(x) = 3(2^x)$
$f(x) = 2(4)^{x-3}$
$f(x) = 2(3^x)$
$f(x) = 2(4^x) + 3$

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": [ "Function Translation", "Vertical Translation", "Exponential Transformations" ], "new_concepts": [ "Horizontal Asymptotes of Exponential Functions" ], "current_concepts": [ "Function Translation", "Vertical Translation", "Exponential Transformations", "Horizontal Asymptotes of Exponential Functions" ] } ## Analyze the standard horizontal asymptote of exponential functions Using the *Exponential Transformations* knowledge point $$ \begin{aligned} &g(x) = a \cdot b^x\ &\lim_{x \to -\infty} g(x) = 0 \quad (\text{for } b > 1)\ &\text{Horizontal Asymptote: } y = 0 \end{aligned} $$ ## Apply vertical translation to find the shifted asymptote Using the *Function Translation* and *Vertical Translation* knowledge points $$ \begin{aligned} &f(x) = a \cdot b^x + k\ &\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (a \cdot b^x + k) = 0 + k = k\ &\text{Horizontal Asymptote: } y = k \end{aligned} $$ ## Evaluate the given options for $k = 3$ Using the *Vertical Translation* knowledge point $$ \begin{aligned} &f(x) = 3(2^x) \implies k = 0 \implies y = 0\ &f(x) = 2(4)^{x-3} \implies k = 0 \implies y = 0\ &f(x) = 2(3^x) \implies k = 0 \implies y = 0\ &f(x) = 2(4^x) + 3 \implies k = 3 \implies y = 3 \end{aligned} $$

(A) $f(x) = 3(2^x)$

(B) $f(x) = 2(4)^{x-3}$

(C) $f(x) = 2(3^x)$

(D) $f(x) = 2(4^x) + 3$

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

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QUESTION IMAGE

Question

Explanation:

Analyze the standard horizontal asymptote of exponential functions

Apply vertical translation to find the shifted asymptote

Evaluate the given options for \(k = 3\)

Answer: