Graphs of Second Degree Polynomials (Quadratic Functions)

Below are several examples of second degree polynomials along with questions and answers at the bottom of the page.

  1. Polynomial: touches the x-axis and opens upward.

    Question 1:

    Why does the parabola open upward and touch the x-axis at only one point?
    Graph of a quadratic polynomial touching x-axis upward
    Figure 1: Quadratic polynomial touching x-axis.
  2. Polynomial: two x-intercepts and opens upward.

    Question 2:

    Why does the parabola intersect the x-axis at two points?
    Graph of quadratic polynomial with two x-intercepts upward
    Figure 2: Quadratic polynomial with two x-intercepts.
  3. Polynomial: two x-intercepts and opens downward.

    Question 3:

    Why does the parabola open downward?
    Graph of quadratic polynomial with two x-intercepts downward
    Figure 3: Quadratic polynomial opening downward.
  4. Polynomial: no x-intercepts and opens upward.

    Question 4:

    Why does the graph have no x-intercepts?
    Graph of quadratic polynomial with no x-intercepts upward
    Figure 4: Quadratic polynomial with no x-intercepts.

Answers to Questions

  1. The parabola opens upward because the leading coefficient in \(f(x) = x^2\) is positive. It touches the x-axis at one point because it has a repeated zero at \(x = 0\).
  2. The parabola intersects the x-axis at two distinct points because it has two distinct zeros, e.g., \(x = 0\) and \(x = 2\).
  3. The parabola opens downward because the leading coefficient in \(f(x) = -2x^2 - 3x + 2\) is negative.
  4. The graph has no x-intercepts because the quadratic \(f(x) = x^2 + 3x + 3\) has no real solutions. The discriminant is \(D = 3^2 - 4(1)(3) = -3 < 0\), so there are no real zeros.

More References on Polynomial Functions