How does the leading coefficient determine the rise and fall of an odd degree polynomial? What is the proof of this statement?
A) If the leading coefficient is positive ( greater than zero ), then the graph falls to the left and rises to the right.
B) If the leading coefficient is negative ( less than zero ), then the graph rises to the left and falls to the right.
Let me show the argument for cubic polynomial: $f(x) = ax^3+bx^2+cx+d$ Write it this way: $$f(x) = x^3 \bigg(a + \frac bx +\frac c{x^2} +\frac d {x^3}\bigg)$$
For high positive values of $x$ the expression inside the bracket is: $a+ $ something going to zero. The expression outside the bracket which is $x^3$ goes to infinity, so depending on the sign of $a$ you get what you want. Clear to see the degree 3 is chosen only for convenience, this argument works for other degrees too.