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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?

    Graph of a second degree polynomial, through the origin and upward
    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?

    Graph of a second degree polynomial.
    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?

    Graph of a second degree polynomial.
    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?

    Graph of a second degree polynomial.
    Figure 4: Graph of a second degree polynomial





Answers to Above Questions
  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.
Polynomial Functions

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