Answer
a. The polynomial function's graph rises to the left end and falls to the right end.
b. Graph $IV$
Work Step by Step
End-Behaviour of a Polynomial
- Polynomial function of odd degree and positive leading coefficient falls to the left end and rises to the right end.
- Polynomial function of odd degree and negative leading coefficient rises to the left end and falls to the right end.
- Polynomial function of even degree and positive leading coefficient rises to the left end and right end.
- Polynomial function of even degree and negative leading coefficient falls to the left end and right end.
a.
In this case, $U(x)=-x^3+2x^2$, the function is of odd degree and negative leading coefficient. therefore, the polynomial function's graph rises to the left end and falls to the right end.
- Using Descartes rules of signs, we can find that the polynomial has $1$ positive zero and $0$ negative zero and has $0$ as a second zero, therefore the polynomial crosses positive $x$ axis 1 time and doesn't cross negative $x$ axis.
b. The graph that matches the description is Graph $IV$
