There are four possible end behaviours for a polynomial function $P(x)$ These are:
$$\begin{aligned}
&\text { as } x \rightarrow \infty, P(x) \rightarrow \infty \text { and as } x \rightarrow-\infty, P(x) \rightarrow \infty \text { or symbolically }(\nwarrow, \nearrow)\\
&\text { as } x \rightarrow \infty, P(x) \rightarrow-\infty \text { and as } x \rightarrow-\infty, P(x) \rightarrow \infty \text { or symbolically }(\nwarrow, \searrow)\\
&\text { as } x \rightarrow \infty, P(x) \rightarrow-\infty \text { and as } x \rightarrow-\infty, P(x) \rightarrow-\infty \text { or symbolically }(\swarrow, \searrow)\\
&\text { as } x \rightarrow \infty, P(x) \rightarrow \infty \text { and as } x \rightarrow-\infty, P(x) \rightarrow-\infty \text { or symbolically }(\nearrow, \nearrow)
\end{aligned}$$
a) By sketching a graph on your GDC, state the type of end behaviour for each of the polynomial functions below.
(i) $P(x)=2 x^{4}-6 x^{3}+x^{2}+4 x-1$
(ii) $P(x)=-2 x^{4}-6 x^{3}+x^{2}+4 x-1$
(iii) $P(x)=-6 x^{3}+x^{2}+4 x-1$
(iv) $P(x)=6 x^{3}+x^{2}-4 x-1$
(v) $P(x)=x^{2}-4 x-1$
(vi) $P(x)=-2 x^{6}+x^{5}+2 x^{4}-3 x^{3}+4 x^{2}-x+1$
(vii) $P(x)=x^{5}+2 x^{4}-x^{3}+x^{2}-x+1$
(viii) $P(x)=-x^{5}+2 x^{4}-x^{3}+x^{2}-x+1$
b) Use your results from a) to write a general statement about how the leading term of a polynomial function, $a_{n} x^{n}$, determines what type of end behaviour the graph of the function will display. Be specific about how the characteristics of the coefficient, $a_{n}$ and the power, $n,$ of the leading term affect the function's end behaviour.