Question about polynomial functions+graphs...?

Since in a polynomial function at numbers near infinity, the highest power of x piece of the function will dominate the rest of the function.

As an example, take f(x) = x^2 + 15x + 1. Note that at low numbers (say 2), you have 4 + 30 + 1. Note that the 30 in the center dominates. However, now take 100. 10000 + 1500 + 1. Note that the x^2 piece is now the larger item. As it goes to infinity, the x^2 piece will completely dominate and the other two pieces are almost ignorable.

Therefore, the determination of whether or not a graph will turn upwards or downward is completely determined by the sign of the highest power of x.

In our example note the x^2 made the number go upward 10000 units. if our equation were f(x) = -x^2 + 15x + 1, we would instead see -10000 + 1500 + 1 ---> steadily going downward!!

This is true in even higher polynomials, but remember to take into account what happens with odd-numbered polynomials!! (they go both up and down) (e.g. -x^3 powered would be going up on the left side, and down on the right!! - just plug in -100 and +100 in for x and see)

In any case, it is all determined by the simple +/- sign of the highest power