which of the following describes the transformation of (g(x) = 3(2)^{-x} + 2) from the parent function (f(x) = 2^x)?

- reflect across the (x)-axis, stretch the graph vertically by a factor of 2, shift 3 units up
- reflect across the (y)-axis, stretch the graph vertically by a factor of 2, shift 3 units up
- reflect across the (y)-axis, stretch the graph vertically by a factor of 3, shift 2 units up
- reflect across the (x)-axis, stretch the graph vertically by a factor of 3, shift 2 units up

Question

which of the following describes the transformation of (g(x) = 3(2)^{-x} + 2) from the parent function (f(x) = 2^x)?

- reflect across the (x)-axis, stretch the graph vertically by a factor of 2, shift 3 units up
- reflect across the (y)-axis, stretch the graph vertically by a factor of 2, shift 3 units up
- reflect across the (y)-axis, stretch the graph vertically by a factor of 3, shift 2 units up
- reflect across the (x)-axis, stretch the graph vertically by a factor of 3, shift 2 units up

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Function Transformations", "Exponential Transformations" ], "current_concepts": [ "Function Transformations", "Exponential Transformations" ] } ## Identify individual transformations $$ \begin{aligned} &f(x) = 2^x \ &g(x) = 3(2)^{-x} + 2 \ &\text{Let } h_1(x) = f(-x) = 2^{-x} \quad \text{(reflection across the } y\text{-axis)} \ &h_2(x) = 3 \cdot h_1(x) = 3(2)^{-x} \quad \text{(vertical stretch by a factor of 3)} \ &g(x) = h_2(x) + 2 = 3(2)^{-x} + 2 \quad \text{(vertical shift up by 2 units)} \end{aligned} $$ ## Match with given options $$ \begin{aligned} &\text{Transformation 1: reflect across the } y\text{-axis} \ &\text{Transformation 2: stretch the graph vertically by a factor of 3} \ &\text{Transformation 3: shift 2 units up} \end{aligned} $$

reflect across the $x$-axis, stretch the graph vertically by a factor of 2, shift 3 units up

reflect across the $y$-axis, stretch the graph vertically by a factor of 2, shift 3 units up

reflect across the $y$-axis, stretch the graph vertically by a factor of 3, shift 2 units up

reflect across the $x$-axis, stretch the graph vertically by a factor of 3, shift 2 units up

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

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QUESTION IMAGE

Question

Explanation:

Identify individual transformations

Match with given options

Answer: