Polynomial functions
The degree and leading coefficient fix the ends; each zero's multiplicity decides whether the graph crosses the axis or touches it.
A worked example
P(x) = x³ − 2x² − 5x + 6. P(1) = 1 − 2 − 5 + 6 = 0, so x − 1 is a factor (the factor theorem). Dividing leaves x² − x − 6 = (x − 3)(x + 2), so the zeros are 1, 3 and −2. Dividing by x − 2 instead leaves a remainder of P(2) = −4.
The mistake people actually make
A factor x + 3 means the zero x = −3. An even multiplicity touches and turns back; an odd one crosses.