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?
a 2x; stretch
b 2; stretch
c 2x; compression
d 2; compression

Answer 1

The original function is \( f(x) = 4^x - 5 \). When this function is transformed to \( f(2x) = 4^{2x} - 5 \), we can rewrite it as:

\[
f(2x) = (4^2)^x - 5 = 16^x - 5
\]

This transformation represents a change in the rate at which the function grows. To analyze this transformation in terms of the parameters:

1. The \( 2x \) in the exponent indicates that the function's input is being compressed horizontally by a factor of \( \frac{1}{2} \) (because it’s a "2" multiplying \( x \)).
2. Since the input \( x \) is multiplied by \( 2 \), it means that the function will grow more rapidly, which will appear as a horizontal compression of the graph.

Given that you are asked to identify a value \( k \) in the context of whether this is a stretch or compression, we can conclude that \( k \) is related to the factor in front of \( x \), which in this case is \( 2 \).

Therefore, analyzing the options provided:

- \( \textbf{a:} \) \( 2x; \) stretch – Incorrect, this would suggest stretching.
- \( \textbf{b:} \) \( 2; \) stretch – Incorrect, this indicates a stretch.
- \( \textbf{c:} \) \( 2x; \) compression – Correct, this indicates a compression.
- \( \textbf{d:} \) \( 2; \) compression – Incorrect in terms of how the notation is applied.

Thus, the correct answer is:

\[
\text{c: } 2x; \text{ compression}
\]

So, the value of \( k \) is \( 2 \), and the transformation results in a horizontal compression.