identify the graph of $y = e^x - 2$.

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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" ] } ## Analyze the parent function and transformation Using the *Exponential Function Graphs* and *Exponential Function Transformations* knowledge points $$ \begin{aligned} &f(x) = e^x \implies \text{horizontal asymptote at } y = 0, \text{ y-intercept at } (0, 1)\ &y = e^x - 2 \implies \text{vertical shift down by 2 units} \end{aligned} $$ ## Determine key features of the transformed graph Using the *Exponential Function Transformations* knowledge point $$ \begin{aligned} &\text{Horizontal Asymptote: } y = -2\ &\text{y-intercept: } (0, e^0 - 2) = (0, -1)\ &\text{x-intercept: } e^x - 2 = 0 \implies x = \ln(2) \approx 0.69 \implies (0.69, 0) \end{aligned} $$ ## Match features to the given options Using the *Exponential Function Graphs* knowledge point $$ \begin{aligned} &\text{Graph 1: Horizontal asymptote at } y = 0, \text{ y-intercept at } (0, 1)\ &\text{Graph 2: Horizontal asymptote at } y = -2, \text{ y-intercept at } (0, -1)\ &\text{Graph 3: Decreasing curve with horizontal asymptote at } y = -2 \end{aligned} $$

(A) Exponential curve with horizontal asymptote at $y = 0$ and y-intercept at $(0, 1)$

(B) Exponential curve with horizontal asymptote at $y = -2$ and y-intercept at $(0, -1)$

(C) Decreasing exponential curve with horizontal asymptote at $y = -2$ and y-intercept at $(0, -1)$

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

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Question

Explanation:

Analyze the parent function and transformation

Determine key features of the transformed graph

Match features to the given options

Answer: