Quadratic Functions and Discriminants

Quadratic Functions and Graphs

  • Function Equations:
    1) y = 2x^2 + 16x + 28
    2) y = 2x^2 - 12x + 16
    3) y = 2x^2 + 12x + 20

  • Vertex of the Parabola:

    • For the function y = 2x^2 + 16x + 28, the vertex is at (-4, -4).

  • Sketching Graphs:

    • Identify key features including the vertex, axis of symmetry, and general shape.

Discriminants and Solutions

  • Finding the Discriminant:

    • Discriminant formula: D = b^2 - 4ac.

    • Determines the number and type of solutions of a quadratic equation.

  • Examples of Quadratic Equations & Discriminants:
    5) For n^2 + 4n - 6 = -10:

    • Rearranged: n^2 + 4n + 4 = 0

    • Discriminant: D = 4^2 - 4(1)(4) = 0 (one rational solution).

    6) For -8b^2 + b + 9 = 9:

    • Rearranged: -8b^2 + b = 0

    • Discriminant: is calculated but seems unclear in this transcript (ensure accurate calculation for solutions).

  • Types of Solutions:

    • If D > 0: Two distinct real solutions

    • If D = 0: One repeated solution (rational)

    • If D < 0: No real solutions (complex solutions).