Question
Write the function rule g(x) after the given transformations of the graph of f(x)equals3x.
reflection in the x-axis; vertical compression by a factor of one ninth
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Part 1
g(x)equals
enter your response here
reflection in the x-axis; vertical compression by a factor of one ninth
Question content area bottom
Part 1
g(x)equals
enter your response here
Answers
GPT-4o mini
To apply the given transformations to the function \( f(x) = 3x \), we will perform the following steps:
1. **Reflection in the x-axis**: Reflecting a function in the x-axis changes the sign of the entire function. Thus, the reflection of \( f(x) = 3x \) will be:
\[
-f(x) = -3x
\]
2. **Vertical compression by a factor of one ninth**: To vertically compress a function by a factor of \( k \), we multiply the function by \( k \). In this case, \( k = \frac{1}{9} \). Therefore, we apply the vertical compression to our reflected function:
\[
g(x) = \frac{1}{9} \cdot (-3x) = -\frac{3}{9}x = -\frac{1}{3}x
\]
Putting it all together, the function rule \( g(x) \) after the transformations is:
\[
g(x) = -\frac{1}{3}x
\]
1. **Reflection in the x-axis**: Reflecting a function in the x-axis changes the sign of the entire function. Thus, the reflection of \( f(x) = 3x \) will be:
\[
-f(x) = -3x
\]
2. **Vertical compression by a factor of one ninth**: To vertically compress a function by a factor of \( k \), we multiply the function by \( k \). In this case, \( k = \frac{1}{9} \). Therefore, we apply the vertical compression to our reflected function:
\[
g(x) = \frac{1}{9} \cdot (-3x) = -\frac{3}{9}x = -\frac{1}{3}x
\]
Putting it all together, the function rule \( g(x) \) after the transformations is:
\[
g(x) = -\frac{1}{3}x
\]