To determine the characteristics of the function [tex]\( f(x) = -(x+4)^5 \)[/tex], we need to analyze its properties step-by-step:
1. End Behavior:
- The function [tex]\( f(x) = -(x+4)^5 \)[/tex] is a polynomial of degree 5.
- A polynomial of odd degree with a negative leading coefficient will have the following end behavior:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- Therefore, the left end of the graph goes up, and the right end goes down.
2. Transformations from the Parent Function [tex]\( f(x) = x^5 \)[/tex]:
- The term [tex]\( (x+4) \)[/tex] indicates a horizontal shift to the left by 4 units.
- The negative sign in front of the expression reflects the graph across the x-axis.
- Therefore, the function [tex]\( f(x) = -(x+4)^5 \)[/tex] is a reflection and a translation to the left of the parent function.
3. Zeros and Relative Extrema:
- The function [tex]\( f(x) \)[/tex] is a polynomial of degree 5, so it may have at most 5 zeros (roots).
- A polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n-1 \)[/tex] relative extrema (maximums or minimums).
- Therefore, a polynomial of degree 5 can have at most 4 relative maxima or minima.
Based on the analysis above, the correct characteristics for the function [tex]\( f(x) = -(x+4)^5 \)[/tex] are:
B. The left end of the graph of the function goes up, and the right end goes down.
C. It is a reflection and a translation to the left of the parent function.
Thus, the corresponding answers are:
- Answer for end behavior: 2
- Answer for transformation: 3
