Question

The exponential equation f(x)=3x−2
has been transformed to 34x−2
. Does this transformation represent a horizontal stretch or a horizontal compression, and what is the value of k?(1 point)
Responses

horizontal compression, k = 4
horizontal compression, k = 4

horizontal stretch, k = 4
horizontal stretch, k = 4

horizontal compression, k = 4x
horizontal compression, k = 4x

horizontal stretch, k = 4x

Answer 1

To analyze the transformation of the function \( f(x) = 3^x - 2 \) to \( f(x) = 3^{4x} - 2 \), we can look at the coefficient of \( x \) in the exponent.

In the original function \( f(x) = 3^x - 2 \), the exponent is simply \( x \). In the transformed function \( f(x) = 3^{4x} - 2 \), the exponent is \( 4x \).

When we compare \( x \) to \( 4x \), we see that the \( x \) value must be multiplied by \( \frac{1}{4} \) in order to reach the original function. This means that we have a transformation affecting the horizontal direction.

- If \( x \) is multiplied by a factor greater than 1 (in this case, 4), it's a horizontal compression.
- If \( x \) is multiplied by a factor between 0 and 1, it's a horizontal stretch.

Therefore, since \( x \) has been scaled by a factor of 4, it represents a horizontal compression.

The value of \( k \) in this scenario is the factor by which \( x \) was compressed, which is \( 4 \).

So, the correct response is:

**horizontal compression, k = 4**.