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

Explanation:

Analyze horizontal transformation

$$ x \to -x \implies \text{reflection across the } y\text{-axis} $$

Analyze vertical transformations

$$
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$$

Combine transformations

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

Answer:

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