To determine how the graph of the parent function [tex]\( y = \frac{1}{x} \)[/tex] is transformed to create the graph of [tex]\( y = -\frac{1}{3x} \)[/tex], let's analyze the given function step by step.
#### Step-by-Step Transformation:
1. Parent Function:
The parent function is [tex]\( y = \frac{1}{x} \)[/tex].
2. Given Function:
The transformed function we are considering is [tex]\( y = -\frac{1}{3x} \)[/tex].
3. Transformations:
- The term [tex]\( \frac{1}{3x} \)[/tex] indicates a horizontal compression by a factor of 3. This is because the denominator has been tripled, which has the effect of compressing the graph horizontally.
- The negative sign in front of the fraction [tex]\( -\frac{1}{3x} \)[/tex] indicates a reflection over the [tex]\( x \)[/tex]-axis. This flips the graph upside down.
Now let's list out the transformations in detail:
- Horizontal Compression by a Factor of 3:
The term [tex]\( \frac{1}{3x} \)[/tex] modifies the horizontal scaling of the graph. For the parent function [tex]\( y = \frac{1}{x} \)[/tex], when [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases. By introducing a factor of 3 in the denominator, the same change in [tex]\( x \)[/tex] will now produce a different change in [tex]\( y \)[/tex]; thus, causing the graph to be horizontally compressed by a factor of 3.
- Reflection over the [tex]\( x \)[/tex]-axis:
The negative sign in [tex]\( -\frac{1}{3x} \)[/tex] means that all positive [tex]\( y \)[/tex]-values of the parent function [tex]\( y = \frac{1}{x} \)[/tex] will now become negative, and all negative [tex]\( y \)[/tex]-values will become positive. This reflection shifts everything symmetrically about the [tex]\( x \)[/tex]-axis.
#### Conclusion:
Given these two transformations, the correct description of how [tex]\( y = -\frac{1}{3x} \)[/tex] is obtained from [tex]\( y = \frac{1}{x} \)[/tex] is that the graph is horizontally compressed by a factor of 3 and reflected over the [tex]\( x \)[/tex]-axis.
Thus, the correct answer is:
It is horizontally compressed by a factor of 3 and reflected over the [tex]\( x \)[/tex]-axis.
