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

Explanation:

Identify the parent function and transformed function

Using the Exponential Function Transformations knowledge point
The parent function is:

$$ f(x) = 2^x $$

The transformed function is:

$$ g(x) = -(2)^{x+4} - 2 $$

Analyze the horizontal translation

Using the Exponential Function Transformations knowledge point

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

Analyze the reflection

Using the Exponential Function Transformations knowledge point

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

Analyze the vertical translation

Using the Exponential Function Transformations knowledge point

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

Combine all transformations

Using the Exponential Function Transformations knowledge point
The transformations are: shift 4 units left, reflect over the \(x\)-axis, and shift 2 units down.

Answer:

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