Question
What are the transformations on the graph of f(x)=1/x if it is replaced by k(x)=−3/4x
It vertically stretches by a factor of 4/3 and reflects across the x-axis.
It vertically shifts up 3/4 units and reflects across the x-axis.
It horizontally stretches by a factor of 4/3 and reflects across the y-axis.
It vertically compresses by a factor of 3/4 and reflects across the x-axis.
It vertically stretches by a factor of 4/3 and reflects across the x-axis.
It vertically shifts up 3/4 units and reflects across the x-axis.
It horizontally stretches by a factor of 4/3 and reflects across the y-axis.
It vertically compresses by a factor of 3/4 and reflects across the x-axis.
Answers
GPT-4o mini
To analyze the transformation from the function \( f(x) = \frac{1}{x} \) to the function \( k(x) = -\frac{3}{4}x \), we need to break down the changes step by step.
1. **Reflection**: The negative sign in front of the \( \frac{3}{4}x \) indicates a reflection across the x-axis.
2. **Vertical Stretch/Compression**: The factor \( \frac{3}{4} \) indicates a vertical compression by a factor of \( \frac{3}{4} \). This is because a factor less than 1 compresses the function vertically.
3. **Horizontal Stretch/Compression/Transformation**: The expression \( x \) in \( -\frac{3}{4}x \) is not altered (there is no division or multiplication by \( x \)), so there is no horizontal stretch or compression from the basic function \( f(x) = \frac{1}{x} \).
Given these transformations, we conclude that:
- The function is **reflected across the x-axis** because of the negative sign.
- The function is **vertically compressed by a factor of \( \frac{3}{4} \)**.
Thus, the correct choice is:
**It vertically compresses by a factor of \( \frac{3}{4} \) and reflects across the x-axis.**
1. **Reflection**: The negative sign in front of the \( \frac{3}{4}x \) indicates a reflection across the x-axis.
2. **Vertical Stretch/Compression**: The factor \( \frac{3}{4} \) indicates a vertical compression by a factor of \( \frac{3}{4} \). This is because a factor less than 1 compresses the function vertically.
3. **Horizontal Stretch/Compression/Transformation**: The expression \( x \) in \( -\frac{3}{4}x \) is not altered (there is no division or multiplication by \( x \)), so there is no horizontal stretch or compression from the basic function \( f(x) = \frac{1}{x} \).
Given these transformations, we conclude that:
- The function is **reflected across the x-axis** because of the negative sign.
- The function is **vertically compressed by a factor of \( \frac{3}{4} \)**.
Thus, the correct choice is:
**It vertically compresses by a factor of \( \frac{3}{4} \) and reflects across the x-axis.**