which function is shown in the graph below?

$y = 2^{x-2} + 3$
$y = 2^{x+2} + 3$
$y = 2^{x-3} + 2$

Question

which function is shown in the graph below?

$y = 2^{x-2} + 3$
$y = 2^{x+2} + 3$
$y = 2^{x-3} + 2$

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** and **Exponential Function Transformations** knowledge points $$ \begin{aligned} &\text{As } x \to -\infty, \text{ the graph approaches the horizontal line } y = 3.\ &\text{Therefore, the vertical shift } k = 3.\ &\text{The function has the form } y = 2^{x-h} + 3. \end{aligned} $$ ## Determine the horizontal translation Using the **Exponential Function Transformations** knowledge point $$ \begin{aligned} &\text{Identify a clear point on the graph: } (0, 7).\ &\text{Substitute } (0, 7) \text{ into } y = 2^{x-h} + 3:\ &7 = 2^{0-h} + 3\ &4 = 2^{-h}\ &2^2 = 2^{-h} \implies h = -2\ &\text{Thus, the equation is } y = 2^{x - (-2)} + 3 \implies y = 2^{x+2} + 3. \end{aligned} $$

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

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

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

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

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QUESTION IMAGE

Question

Explanation:

Identify the horizontal asymptote

Determine the horizontal translation

Answer: