Question

question 1 of 10
what are the characteristics of the function $f(x) = -(x + 4)^5$?

a. it has 4 zeros and at most 5 relative maximums or minimums.

b. the left end of the graph of the function goes down, and the right end goes up.

c. it has 5 zeros and at most 4 relative maximums or minimums.

d. the left end of the graph of the function goes up, and the right end goes down.

e. it is a vertical stretch and a translation to the left of the parent function.

f. it is a reflection and a translation to the left of the parent function.

Explanation:

Step1: Analyze zeros

Set $f(x)=0$: $-(x+4)^5=0 \Rightarrow x=-4$ (1 zero, multiplicity 5). So A/C wrong.

Step2: Analyze end behavior

Degree=5 (odd), leading coefficient=-1 (negative). Left end: $x\to-\infty$, $f(x)\to\infty$ (up); Right end: $x\to\infty$, $f(x)\to-\infty$ (down). So B wrong, D correct.

Step3: Analyze transformations

Parent function: $y=x^5$. $f(x)=-(x+4)^5$: $x+4$ (left translation 4), negative sign (reflection over x-axis). No vertical stretch. So E wrong, F correct.

Answer:

D. The left end of the graph of the function goes up, and the right end goes down., F. It is a reflection and a translation to the left of the parent function.