Question

The equation f(x)=4^x-5 is transformed to 4^2x-5. Identify the value of k Does the graph show a stretch or a compression
2; stretch
2: compression
2x; compression
2x; stretch

Answer 1

The transformation you are describing involves the function \( f(x) = 4^x - 5 \) being changed to \( f(2x) = 4^{2x} - 5 \).

To analyze this transformation:

1. **Comparing the original and transformed functions**:
- The original is \( f(x) = 4^x - 5 \).
- The transformed function is \( f(2x) = 4^{2x} - 5 \).

2. **Identifying the effect of the transformation**:
- The transformation from \( f(x) \) to \( f(2x) \) introduces a horizontal compression.
- Specifically, the base \( 4^x \) changes to \( 4^{2x} \), which corresponds to the geometric interpretation of scaling the input \( x \) by a factor of \( \frac{1}{2} \), leading to points on the graph occurring faster (which is a compression).

3. **Value of \( k \)**:
- In the context of the transformation, we can say that the function undergoes a compression by a factor of \( 2 \) because the \( 2x \) argument compresses the graph horizontally.

Based on this analysis:

- The value of \( k \) is \( 2 \).
- The graph shows a **compression** (option "2x; compression").

Therefore, the final answer is:
- Value of \( k \): **2**
- Type of transformation: **compression**.