Question

which of the following describes the transformations of (g(x) = -(2)^{x+4} - 2) from the parent function (f(x) = 2^x)?

• shift 4 units left, reflect over the (y)-axis, shift 2 units down
• shift 4 units right, reflect over the (y)-axis, shift 2 units down
• shift 4 units left, reflect over the (x)-axis, shift 2 units down
• shift 4 units right, reflect over the (x)-axis, shift 2 units down

Explanation:

Analyze horizontal shift

$$ x \to x + 4 \implies \text{shift 4 units left} $$

Analyze reflection

$$ y \to -y \implies \text{reflect over the } x\text{-axis} $$

Analyze vertical shift

$$ y \to y - 2 \implies \text{shift 2 units down} $$

Answer:

• shift 4 units left, reflect over the \(y\)-axis, shift 2 units down
• shift 4 units right, reflect over the \(y\)-axis, shift 2 units down
• shift 4 units left, reflect over the \(x\)-axis, shift 2 units down (Correct answer)
• shift 4 units right, reflect over the \(x\)-axis, shift 2 units down