Answer :
To determine if the order of transformations matters when graphing the function [tex]\( y = -\cot(x) - 1 \)[/tex], let's go through the steps in detail.
### Step-by-Step Analysis
1. Original Function Analysis:
The original function given is [tex]\( y = \cot(x) \)[/tex].
2. Transformation Details:
We need to reflect this graph about the [tex]\( x \)[/tex]-axis and then translate it 1 unit down. We will analyze both possible orders of doing these operations.
#### Case 1: Reflecting First, Then Translating
1. Reflecting about the [tex]\( x \)[/tex]-axis:
Reflecting [tex]\( y = \cot(x) \)[/tex] about the [tex]\( x \)[/tex]-axis changes the function to:
[tex]\[ y = -\cot(x) \][/tex]
2. Translating 1 Unit Down:
After reflecting, we translate the resulting graph 1 unit down by subtracting 1 from it:
[tex]\[ y = -\cot(x) - 1 \][/tex]
This gives us the desired function [tex]\( y = -\cot(x) - 1 \)[/tex].
#### Case 2: Translating First, Then Reflecting
1. Translating 1 Unit Down:
Translating [tex]\( y = \cot(x) \)[/tex] 1 unit down:
[tex]\[ y = \cot(x) - 1 \][/tex]
2. Reflecting about the [tex]\( x \)[/tex]-axis:
Reflecting the new function [tex]\( y = \cot(x) - 1 \)[/tex] about the [tex]\( x \)[/tex]-axis changes the function to:
[tex]\[ y = - (\cot(x) - 1) = -\cot(x) + 1 \][/tex]
In this case, we end up with the function [tex]\( y = -\cot(x) + 1 \)[/tex], which is not the same as [tex]\( y = -\cot(x) - 1 \)[/tex].
### Conclusion
Through the analysis, we see that:
- Reflecting about the [tex]\( x \)[/tex]-axis first and then translating 1 unit down produces [tex]\( y = -\cot(x) - 1 \)[/tex].
- Translating 1 unit down first and then reflecting about the [tex]\( x \)[/tex]-axis produces [tex]\( y = -\cot(x) + 1 \)[/tex].
Since these two final functions are not equivalent, the order of transformations does matter.
### Correct Conclusion
Therefore, Eve should reflect the graph of the function [tex]\( y = \cot(x) \)[/tex] about the [tex]\( x \)[/tex]-axis before it is translated 1 unit down.
The correct answer is:
Yes, it matters. The graph of the function [tex]\( y = \cot(x) \)[/tex] should be reflected about the [tex]\( x \)[/tex]-axis before it is translated 1 unit down.
### Step-by-Step Analysis
1. Original Function Analysis:
The original function given is [tex]\( y = \cot(x) \)[/tex].
2. Transformation Details:
We need to reflect this graph about the [tex]\( x \)[/tex]-axis and then translate it 1 unit down. We will analyze both possible orders of doing these operations.
#### Case 1: Reflecting First, Then Translating
1. Reflecting about the [tex]\( x \)[/tex]-axis:
Reflecting [tex]\( y = \cot(x) \)[/tex] about the [tex]\( x \)[/tex]-axis changes the function to:
[tex]\[ y = -\cot(x) \][/tex]
2. Translating 1 Unit Down:
After reflecting, we translate the resulting graph 1 unit down by subtracting 1 from it:
[tex]\[ y = -\cot(x) - 1 \][/tex]
This gives us the desired function [tex]\( y = -\cot(x) - 1 \)[/tex].
#### Case 2: Translating First, Then Reflecting
1. Translating 1 Unit Down:
Translating [tex]\( y = \cot(x) \)[/tex] 1 unit down:
[tex]\[ y = \cot(x) - 1 \][/tex]
2. Reflecting about the [tex]\( x \)[/tex]-axis:
Reflecting the new function [tex]\( y = \cot(x) - 1 \)[/tex] about the [tex]\( x \)[/tex]-axis changes the function to:
[tex]\[ y = - (\cot(x) - 1) = -\cot(x) + 1 \][/tex]
In this case, we end up with the function [tex]\( y = -\cot(x) + 1 \)[/tex], which is not the same as [tex]\( y = -\cot(x) - 1 \)[/tex].
### Conclusion
Through the analysis, we see that:
- Reflecting about the [tex]\( x \)[/tex]-axis first and then translating 1 unit down produces [tex]\( y = -\cot(x) - 1 \)[/tex].
- Translating 1 unit down first and then reflecting about the [tex]\( x \)[/tex]-axis produces [tex]\( y = -\cot(x) + 1 \)[/tex].
Since these two final functions are not equivalent, the order of transformations does matter.
### Correct Conclusion
Therefore, Eve should reflect the graph of the function [tex]\( y = \cot(x) \)[/tex] about the [tex]\( x \)[/tex]-axis before it is translated 1 unit down.
The correct answer is:
Yes, it matters. The graph of the function [tex]\( y = \cot(x) \)[/tex] should be reflected about the [tex]\( x \)[/tex]-axis before it is translated 1 unit down.
