SAT Math Section 12: Quadratics

Importance of Quadratics

• Extremely important and common in the SAT.
• Tested in many variations.
• Requires in-depth knowledge and familiarity.

Definition and Identification

• Quadratic Function: Polynomial of degree 2.
• Standard Form: f(x) = ax^2 + bx + c
  • Involves two variables (e.g., y and x, w and z, p(n) and n).
  • One variable is squared, and the other is of the first degree.
  • Example: h(a) = a^2 + 2a, where a is squared and h (or y) is to the first degree.
• Identifying a quadratic relationship requires recognizing one variable squared and another to the first degree.

Forms of Quadratic Functions

1. Standard Form

• y = ax^2 + bx + c
  • Can also be written as f(x) = ax^2 + bx + c

2. Vertex Form

• y = a(x - h)^2 + k
  • Alternate form: y - k = a(x - h)^2

3. Intercept (Factored) Form

• y = a(x - p)(x - q)

Recognizing and Understanding Forms

• Recognize the forms by their structure.
• Understand why they are named so (e.g., Vertex form allows identification of the vertex).
• Know what information each form provides (e.g., roots, intercepts, vertex).

Graphs of Quadratic Functions

• Graphs of quadratics are parabolas (U-shaped curves).

Important Features and Points on a Parabola

1. X-Intercepts (Roots or Zeros):

• Where the parabola intersects the x-axis.

2. Y-Intercept:

• Where the parabola intersects the y-axis.

3. Vertices (Singular: Vertex):

• Points where the parabola changes direction (turning points).
  • Classified as minimum (C) or maximum (D).
  • Important for understanding the function's minimum or maximum values.

4. Axis of Symmetry:

• Vertical line through the vertex.
  • Parabolas are symmetric about this line.
  • Points equidistant horizontally from the axis of symmetry are mirror images.

Summary of Key Concepts

• Symmetry and reflection about the axis of symmetry.
• Minimum/maximum values at the vertex.
• The uniform shape and identifying the features of the graph.
• Relations between different forms and the graphical representation.