Quadratic Functions and Parabolas

Quadratic Functions

• A quadratic function is expressed as y = ax² + bx + c, where a ≠ 0.
• Graphing a quadratic function results in a U-shaped figure known as a parabola.

Parabola Characteristics

• The vertex is the highest or lowest point on the parabola.
• An axis of symmetry divides the parabola into two mirror-image halves; for y = x² and y = ax², symmetry exists around the y-axis.
• A parabola opens upwards if a > 0 and downwards if a < 0.

Vertex and Axis of Symmetry

• The x-coordinate of the vertex can be found using x = -b/2a.
• The axis of symmetry is the vertical line x = -b/2a.

Forms of Quadratic Functions

• Vertex form: y = a(x-h)² + k, where (h, k) is the vertex.
• Intercept form: y = a(x-p)(x-q), where p and q represent x-intercepts.

Graph Characteristics

• In vertex form, the vertex is (h, k) and the axis of symmetry is x = h.
• The x-intercepts p and q are located on the x-axis, with the axis of symmetry situated midway between these two points.

Binomials and Trinomials

• Binomials are algebraic expressions consisting of two terms, e.g., x + 3, x + 5.
• Trinomials consist of three terms, e.g., x² + 8x + 15.

Factoring Methods

• Factoring involves expressing a trinomial as a product of binomials: x² + bx + c = (x + m)(x + n), where the sum m + n = b and the product mn = c.

Special Factoring Cases

• Difference of Two Squares formula: a² - b² = (a + b)(a - b).
• Perfect Square Trinomial formulas:
  • a² + 2ab + b² = (a + b)²
  • a² - 2ab + b² = (a - b)²

Standard Form of Quadratic Equations

• A quadratic equation can be represented as ax² + bx + c = 0, known as the standard form, where a ≠ 0.

Zero Product Property

• If the left side of the equation ax² + bx + c = 0 is factorable, the Zero Product Property applies: If AB = 0, then A = 0 or B = 0.