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How is the graph of the parent function [tex]$y=\frac{1}{x}$[/tex] transformed to create the graph of [tex]$y=-\frac{1}{3x}$[/tex]?

A. It is horizontally stretched by a factor of 3 and reflected over the [tex]$y$[/tex]-axis.
B. It is translated 3 units down and reflected over the [tex]$x$[/tex]-axis.
C. It is horizontally compressed by a factor of 3 and reflected over the [tex]$x$[/tex]-axis.
D. It is translated 3 units down and reflected over the [tex]$y$[/tex]-axis.

Sagot :

GinnyAnsweridn
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.
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