Quadratic Relations Cheat Sheet

Quadratic relations model parabolic paths and relationships, characterized by an x² term and constant second differences. They are essential for understanding motion, optimization, and shape in various real-world applications.

Core Principles

• The graph of a quadratic relation is a parabola.
• Quadratic relations have a constant second difference when analyzed in a table of values.
• The general form of a quadratic relation is y = ax² + bx + c.
• The vertex form of a quadratic relation is y = a(x - h)² + k.
• The value of 'a' determines the parabola's orientation (upward/downward) and vertical stretch/compression.
• The value of 'k' determines the vertical shift of the parabola.
• The value of 'h' determines the horizontal shift of the parabola.
• The vertex of the parabola is at (h, k).

Action Steps

• Identify if a relation is quadratic by checking for an x² term or constant second differences.
• Determine the vertex (h, k) from the vertex form equation.
• Analyze the 'a' value to understand the parabola's shape and direction.
• Analyze the 'k' value to understand the vertical shift.
• Analyze the 'h' value to understand the horizontal shift.
• Use the vertex and other points to sketch the parabola.
• Apply quadratic models to real-world scenarios like projectile motion or optimization problems.

Formulas

• Standard Form: $y = ax^2 + bx + c$
• Vertex Form: $y = a(x - h)^2 + k$
• Vertex Coordinates: $(h, k)$

Key Terms

• Parabola: A symmetrical U-shaped curve that is the graph of a quadratic relation.
• Vertex: The highest or lowest point on a parabola; the turning point.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror images, passing through the vertex.
• Maximum: The highest point on a parabola that opens downward (the y-coordinate of the vertex).
• Minimum: The lowest point on a parabola that opens upward (the y-coordinate of the vertex).
• Vertex Form: The form y = a(x - h)² + k, which clearly shows the vertex (h, k) and the stretch/compression factor 'a'.
• Mathematical Model: A mathematical description of a real-world situation using equations, graphs, or tables.

Real World Examples

• Projectile Motion: Modeling the path of a thrown ball, a kicked football, or a launched rocket.
• Optimization Problems: Finding the maximum area, revenue, or height, or the minimum cost or distance.
• Engineering and Design: Designing suspension bridges, satellite dishes, or car headlights.
• Computer Graphics: Creating realistic motion for characters and objects in video games.

Timeline

• 3000 B.C.E.: Ancient Babylonians studied quadratic relations in the context of farming.
• 17th Century: Galileo Galilei showed that projectiles travel in parabolic arcs.
• 3rd Century B.C.E.: Archimedes discovered properties of parabolic reflectors.
• 2003: Quadratic relations were the subject of a debate in the UK House of Commons.

People

• Archimedes: Discovered properties of parabolic reflectors.
• Galileo Galilei: Showed that projectiles travel in parabolic arcs.