How can I find out how a polynomial graph opens up or down

A rather vague question!

However, if I am not mistaken you simply want the abridged answer.

Look at the highest x term => if even and coefficient is positive => opens up; if coefficient is negative, it opens down.

Also, so many movements may happen in between....but your question does not specify that.

Look at the highest order term and its coefficient. The highest order term will dominate all the others for |x| very large, and will ultimately determine how the graph looks at the ends. (In between it can do a lot of things, especially for high degree polynomials.)

If the power is even, it will open up if the coefficient is positive, and down if the coefficient is negative. Again, it might go up and down in the middle, but I'm presuming that the behavior for |x| large that determine whether we'll say that it opens up or down.

If the power is odd and the coefficient is positive, it will go from negative infinity for large negative x, and positive infinity for large positive x. If the coefficient is negative, it will go from positive infinity for large negative x, and to negative infinity for large positive x.

By using the slope. Positive slope means line goes up diagonally. Negative slope means line goes down diagonally.

By looking at the sign of the function