which function is graphed on the right?
$ y = 2^{x+3} - 2 $
$ y = 2^{x-3} + 2 $
$ y = 2^{x-2} + 3 $
$ y = 2^{x-2} - 3 $

Question

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": [ "Exponential Function Graphs", "Exponential Function Transformations" ], "new_concepts": [], "current_concepts": [ "Exponential Function Graphs", "Exponential Function Transformations" ] } ## Identify the horizontal asymptote Using the *Exponential Function Graphs* knowledge point $$ \begin{aligned} &\text{As } x \to -\infty, \text{ the graph approaches } y = 3.\ &\text{Therefore, the horizontal asymptote is } y = 3. \end{aligned} $$ ## Determine the vertical shift Using the *Exponential Function Transformations* knowledge point $$ \begin{aligned} &\text{An exponential function of the form } y = a \cdot 2^{x-h} + k \text{ has a horizontal asymptote at } y = k.\ &\text{Since the asymptote is } y = 3, \text{ we have } k = 3. \end{aligned} $$ ## Identify key points on the graph Using the *Exponential Function Graphs* knowledge point $$ \begin{aligned} &\text{The graph passes through the point } (2, 4).\ &\text{The graph also passes through the point } (3, 5). \end{aligned} $$ ## Find the horizontal translation Using the *Exponential Function Transformations* knowledge point $$ \begin{aligned} &y = 2^{x-h} + 3\ &4 = 2^{2-h} + 3 \implies 1 = 2^{2-h} \implies 2-h = 0 \implies h = 2\ &\text{Thus, the equation is } y = 2^{x-2} + 3. \end{aligned} $$

(A) $y = 2^{x+3} - 2$

(B) $y = 2^{x-3} + 2$

(C) $y = 2^{x-2} + 3$

(D) $y = 2^{x-2} - 3$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Function Graphs" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the horizontal asymptote

Determine the vertical shift

Identify key points on the graph

Find the horizontal translation

Answer: