Answer :
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.
