Question
The exponential equation f (x) = 32 has been transformed to 34z-2. Does this transformation represent a horizontal stretch or a horizontal compression, and what is the value of k? (1 point)
horizontal compression, k = 4
horizontal stretch, k = 4
horizontal compression, k =.4x
horizontal stretch, k = 4x
horizontal compression, k = 4
horizontal stretch, k = 4
horizontal compression, k =.4x
horizontal stretch, k = 4x
Answers
GPT-4o mini
To analyze the transformation from the function \( f(x) = 32 \) to \( f(z) = 34z - 2 \), we need to understand how to interpret the transformations involving the variable \( z \).
First, we can rewrite the transformed function in the form of the original function:
1. The base of the function in question is 3, indicated by \( 34z - 2 \), suggesting we look at the transformation of \( f(x) = 3^x \) instead of \( f(x) = 32 \).
2. If the transformation shifts horizontally and involves a scaling factor, it can usually be discussed in terms of changes related to the variable \( z \).
Given \( f(z) = 34z - 2 \), we have an expression that involves a modification of \( z \). To determine whether this represents a horizontal stretch or compression, we should typically look at the coefficient of \( z \).
The relevant transformation of the variable would involve a function of the form \( f(kz) \). Here, the function is \( 34z \). To see whether it represents stretch or compression:
- If \( k > 1 \), it generally indicates a horizontal compression.
- If \( 0 < k < 1 \), it indicates a horizontal stretch.
In the expression \( 34z \), we can identify \( k = 4 \) because you can factor it as \( f(z) = 3^{4z} - 2 \).
Thus, since \( k = 4 \) (which is greater than 1), we see this implies a **horizontal compression**.
Therefore, the correct option is:
**horizontal compression, k = 4**.
First, we can rewrite the transformed function in the form of the original function:
1. The base of the function in question is 3, indicated by \( 34z - 2 \), suggesting we look at the transformation of \( f(x) = 3^x \) instead of \( f(x) = 32 \).
2. If the transformation shifts horizontally and involves a scaling factor, it can usually be discussed in terms of changes related to the variable \( z \).
Given \( f(z) = 34z - 2 \), we have an expression that involves a modification of \( z \). To determine whether this represents a horizontal stretch or compression, we should typically look at the coefficient of \( z \).
The relevant transformation of the variable would involve a function of the form \( f(kz) \). Here, the function is \( 34z \). To see whether it represents stretch or compression:
- If \( k > 1 \), it generally indicates a horizontal compression.
- If \( 0 < k < 1 \), it indicates a horizontal stretch.
In the expression \( 34z \), we can identify \( k = 4 \) because you can factor it as \( f(z) = 3^{4z} - 2 \).
Thus, since \( k = 4 \) (which is greater than 1), we see this implies a **horizontal compression**.
Therefore, the correct option is:
**horizontal compression, k = 4**.