How would I find the zeros of this function?
$\displaystyle f(x)=-2x^3(x+5)^4(x-3)^2$
And would the end behavior be Rise Left, Falls Right? Would the degree be all the degrees added or the highest degree in this function?
The zero's just set each term equal to zero and solve, by the zero product principle.
For end behavior, we add all of the exponents which would give us nine, so an odd function has opposite end behavior, and since it is negative it will come in on the left from the top and leave on the right in the bottom.
Hope that helps.
The graph is correct - since it's a 9th degree polynomial with a negative sign out front, it will come in at the top of the graph on the left and "leave" on the right at the bottom.
The idea is that a graph's end behavior can be determined by putting the function into one of four categories, based on its degree and whether its leading term is positive or negative.
1) Even degree, positive leading term - behaves like y = x^2, i.e., the y-values go to positive infinity in both directions.
2) Even degree, negative leading term - behaves like y = -(x^2), i.e., the y-values go to negative infinity in both directions.
3) Odd degree, positive leading term - behaves like y = x, i.e., the y-values go to negative infinity as x goes to negative infinity, and the y-values go to positive infinity as x goes to positive infinity.
4) Odd degree, negative leading term - behaves like y = -x, i.e., the y-values go to positive infinity as x goes to negative infinity, and the y-values go to negative infinity as x goes to positive infinity.
New-ish textbooks seems to use this "Rise Left, Fall Right" (and the other three combinations therein) instead of defining it in any sort of mathematical terms (as I did above). Personally, I find the "Rise Left" notion confusing - does that mean it's rising as we GO to the left, or that we should "Rise" as we read the graph from left to right? (These different interpretations result in two completely different graphs.)
So I prefer the mathematical precision of talking about, basically, limits.