Unit 3 - Quadratics - Mr. Scott's Math Class

Curriculum Outcomes

Relations and Functions

• Develop algebraic and graphical reasoning through the study of relations.
• Analyze quadratic functions of the forms:
  • y = a(x - p)² + q:
    • Determine the vertex, domain and range, direction of opening, axis of symmetry, x- and y-intercepts.
  • y = ax² + bx + c:
    • Identify characteristics such as vertex, domain and range, direction of opening, axis of symmetry, x- and y-intercepts.
    • Solve related problems.
• Solve problems involving quadratic equations.

Key Concepts

Quadratics Overview

• Quadratics are polynomials with a degree (highest exponent) of 2.
• The graph of a quadratic is a parabola, which can open upwards (U-shaped) or downwards (∩-shaped).
• X-intercepts (roots) are points where the graph crosses the x-axis.

Forms of Quadratic Equations

• Vertex Form:
  • Equation: y = a(x - p)² + q
  • Vertex: (p, q) is the lowest/highest point on the graph.
• Standard Form:
  • Equation: y = ax² + bx + c
  • The y-intercept is the value of c.
  • The vertex can be found using x = -b/2a and substituting back to find the y-coordinate.

Converting Between Forms

• From vertex to standard form: Expand (multiply out) the equation.
• From standard to vertex form: Complete the square or use x = -b/2a to find the vertex.
• The "a" value remains constant across both forms.

Finding Roots of a Quadratic

• Set the quadratic equation to equal 0.
• Methods to find roots include:
  • Graphing calculator
  • Factoring
  • Vertex form rearrangement
  • Quadratic formula
• Discriminant (b² - 4ac) determines the number of roots:

  • 0: Two solutions

  • = 0: One solution
  • < 0: No solutions

Resources

Video Tutorials

• Vertex form explanation and examples.
• Standard form explanations.
• Completing the square for form conversion.
• Finding roots through graphing and factoring.
• Quadratic formula usage.
• Factoring method, also known as "Adapted Sum/Product Rule" or "Decomposition."

Note

• Links to video tutorials and additional resources are available for further exploration and practice.