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.

1. 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.
   
2. 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
   
3. 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
   
4. 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
   

1. 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.
   
2. The parabola cuts the x axis at two distinct points because it has two distinct zerso at x = 0 and x = 2.
   
3. The parabola opens downward because the leading coefficient in f(x) = -2x² - 3x + 2 is negative.
   
4. 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