Question

The equation f(x)=4x−5
is transformed to 4^2x−5
. Identify the value of k. Does the graph show a stretch or a compression?(1 point)
Responses

2x
; stretch
2 x ; stretch

2x
; compression
2 x ; compression

2; stretch
2; stretch

2; compression

Answer 1

To analyze the transformation of the function \( f(x) = 4x - 5 \) to \( f(x) = 4^{2x} - 5 \), we need to consider the nature of the exponential function involved.

1. The original function is a linear function, while the transformed function \( 4^{2x} - 5 \) is an exponential function.
2. The transformation from \( 4x \) to \( 4^{2x} \) indicates that we are dealing with an exponential growth function where the exponent \( 2x \) effectively scales the input \( x \).

### Identifying the Value of \( k \):
The term \( 2x \) can be rewritten to show that it is \( kx \) where \( k = 2 \).

### Stretch or Compression:
In terms of transformations:
- If the function’s graph of the exponential \( a^{kx} \) (where \( k > 1 \)) expands vertically, we call this a **stretch**.
- If \( 0 < k < 1 \), it compresses (flattens) the graph.

Since \( k = 2 \) (which is greater than 1), the graph shows a **stretch**.

### Conclusion:
The value of \( k \) is \( 2 \), and the graph shows a **stretch**.

Thus, the correct response is:
**2; stretch**