The graphs of several second degree polynomials are shown along with questions and answers
at the bottom of the page.

Polynomial of a second degree polynomial: touches the x axis and upward.
Question 1: Why does the parabola open upward? Why does it touch the x axis at one point only?
Figure 1: Graph of a second degree polynomial.

Polynomial of a second degree polynomial: two x intercepts and upward.
Question 2: Why does the parabola cut the x axis at two distinct points?
Figure 2: Graph of a second degree polynomial

Polynomial of a second degree polynomial: two x intercepts and downward
Question 3: Why does the parabola open downward?
Figure 3: Graph of a second degree polynomial

Polynomial of a second degree polynomial: no x intercept and upward
Question 4: Why does the graph have no xintercepts?
Figure 4: Graph of a second degree polynomial
Answers to Above Questions

The parabola opens upward because the leading coefficient in f(x) = x^{2} is positive. The parabola touches the x axis because it has a repeated zero at x = 0.
 The parabola cuts the x axis at two distinct points because it has two distinct zerso at x = 0 and x = 2.

The parabola opens downward because the leading coefficient in f(x) = 2x^{2}  3x + 2 is negative.

The graph has no x intercepts because f(x) = x^{2} + 3x + 3 has no zeros. Find the discriminant D of x^{2} + 3x + 3; D = 9  12 = 3. Since the discriminant is negative, then x^{2} + 3x + 3 = 0 has no solution.
More References and Links to Polynomial Functions
Polynomial Functions 