Graphs of Second Degree Polynomials

 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 x-intercepts? Figure 4: Graph of a second degree polynomial Answers to Above Questions The parabola opens upward because the leading coefficient in f(x) = x2 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) = -2x2 - 3x + 2 is negative. The graph has no x intercepts because f(x) = x2 + 3x + 3 has no zeros. Find the discriminant D of x2 + 3x + 3; D = 9 - 12 = -3. Since the discriminant is negative, then x2 + 3x + 3 = 0 has no solution. More references and links to polynomial functions. Polynomial Functions