Join the growing community of curious minds on IDNLearn.com. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
Sagot :
To determine the characteristics of the function [tex]\( f(x) = 2(x-4)^5 \)[/tex], let's analyze its properties step by step.
### 1. Zeros of the Function
The zero of the function occurs when the function equals zero:
[tex]\[ f(x) = 0 \][/tex]
[tex]\[ 2(x-4)^5 = 0 \][/tex]
Dividing both sides by 2:
[tex]\[ (x-4)^5 = 0 \][/tex]
Taking the fifth root of both sides:
[tex]\[ x-4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, the only zero of the function is [tex]\( x = 4 \)[/tex].
### 2. Transformation
The function [tex]\( f(x) = 2(x-4)^5 \)[/tex] is a transformation of the parent function [tex]\( x^5 \)[/tex]:
- It is translated 4 units to the right because of the term [tex]\( (x-4) \)[/tex].
- It is vertically stretched by a factor of 2 because of the coefficient 2 in front.
### 3. End Behavior
End behavior describes how the function behaves as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex]:
- For large positive [tex]\( x \)[/tex] (i.e., [tex]\( x \rightarrow +\infty \)[/tex]): [tex]\( (x-4)^5 \)[/tex] is positive and therefore [tex]\( f(x) \)[/tex] is positive and will go up.
- For large negative [tex]\( x \)[/tex] (i.e., [tex]\( x \rightarrow -\infty \)[/tex]): [tex]\( (x-4)^5 \)[/tex] is negative and therefore [tex]\( f(x) \)[/tex] is negative and will go down.
This implies:
- The left end of the graph goes down.
- The right end of the graph goes up.
### 4. Relative Maximums or Minimums
Since the function is a 5th-degree polynomial, it can have at most 4 relative extrema (either maximums or minimums). This is a property of polynomial functions where the number of possible relative extrema is one less than the degree.
### Summary of Characteristics
- Zeros: [tex]\( x = 4 \)[/tex] (only one zero).
- Translation: 4 units to the right.
- Vertical Stretch: by a factor of 2.
- Left End Behavior: Goes down.
- Right End Behavior: Goes up.
- Relative Maximums or Minimums: At most 4.
### Evaluating the Given Options
- Option A: Incorrect, as the function has 1 zero, not 4.
- Option B: Incorrect, as the function has 1 zero, not 5.
- Option C: Incorrect, as it's not a reflection, only a stretch and translation.
- Option D: Incorrect, as the left end goes down and the right end goes up.
- Option E: Correct, it is a vertical stretch and a translation to the right of the parent function.
- Option F: Correct, the left end of the graph of the function goes down, and the right end goes up.
### Conclusion
The correct options are:
- E. It is a vertical stretch and a translation to the right of the parent function.
- F. The left end of the graph of the function goes down, and the right end goes up.
### 1. Zeros of the Function
The zero of the function occurs when the function equals zero:
[tex]\[ f(x) = 0 \][/tex]
[tex]\[ 2(x-4)^5 = 0 \][/tex]
Dividing both sides by 2:
[tex]\[ (x-4)^5 = 0 \][/tex]
Taking the fifth root of both sides:
[tex]\[ x-4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, the only zero of the function is [tex]\( x = 4 \)[/tex].
### 2. Transformation
The function [tex]\( f(x) = 2(x-4)^5 \)[/tex] is a transformation of the parent function [tex]\( x^5 \)[/tex]:
- It is translated 4 units to the right because of the term [tex]\( (x-4) \)[/tex].
- It is vertically stretched by a factor of 2 because of the coefficient 2 in front.
### 3. End Behavior
End behavior describes how the function behaves as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex]:
- For large positive [tex]\( x \)[/tex] (i.e., [tex]\( x \rightarrow +\infty \)[/tex]): [tex]\( (x-4)^5 \)[/tex] is positive and therefore [tex]\( f(x) \)[/tex] is positive and will go up.
- For large negative [tex]\( x \)[/tex] (i.e., [tex]\( x \rightarrow -\infty \)[/tex]): [tex]\( (x-4)^5 \)[/tex] is negative and therefore [tex]\( f(x) \)[/tex] is negative and will go down.
This implies:
- The left end of the graph goes down.
- The right end of the graph goes up.
### 4. Relative Maximums or Minimums
Since the function is a 5th-degree polynomial, it can have at most 4 relative extrema (either maximums or minimums). This is a property of polynomial functions where the number of possible relative extrema is one less than the degree.
### Summary of Characteristics
- Zeros: [tex]\( x = 4 \)[/tex] (only one zero).
- Translation: 4 units to the right.
- Vertical Stretch: by a factor of 2.
- Left End Behavior: Goes down.
- Right End Behavior: Goes up.
- Relative Maximums or Minimums: At most 4.
### Evaluating the Given Options
- Option A: Incorrect, as the function has 1 zero, not 4.
- Option B: Incorrect, as the function has 1 zero, not 5.
- Option C: Incorrect, as it's not a reflection, only a stretch and translation.
- Option D: Incorrect, as the left end goes down and the right end goes up.
- Option E: Correct, it is a vertical stretch and a translation to the right of the parent function.
- Option F: Correct, the left end of the graph of the function goes down, and the right end goes up.
### Conclusion
The correct options are:
- E. It is a vertical stretch and a translation to the right of the parent function.
- F. The left end of the graph of the function goes down, and the right end goes up.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.
