Answer
a. The polynomial function's graph rises to the left end and rises to the right end.
b. Graph $II$
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, $S(x)=\frac{1}{2}x^6-2x^4$, the function is of even degree and positive leading coefficient. therefore, the polynomial function's graph rises to the left end and rises to the right end.
- Using Descartes rules of signs, we can find that the polynomial has $1$ positive zero and $1$ negative zero and $0$ as a third zero, therefore the polynomial crosses positive $x$ axis $1$ times and negative $x$ axis $1$ times
b. The graph that matches the description is Graph $II$
