Lesson: Converting Quadratic Functions Between Standard and Vertex Form

Overview

This lesson covers how to convert a quadratic function from standard form to vertex form and vice versa. The key techniques involve completing the square and using the FOIL method for expansion.

Converting Standard Form to Vertex Form

1. Complete the Square Method

   • Given quadratic: ax^2 + bx + c
   • To complete the square:
     • Take half of the coefficient of the linear term (b), square it, and add it to both sides of the equation.
     • Example: For x^2 + 6x, half of 6 is 3, so add 3^2 = 9 to both sides.
     • Adjust the equation: x^2 + 6x + 9 and -9 to balance the equation.

2. Factoring the Perfect Square Trinomial

   • Forms a perfect square trinomial: (x + 3)^2
   • Resulting vertex form: (x + 3)^2 - 14

3. Finding the Vertex

   • From vertex form (x + 3)^2 - 14, the vertex is at (-3, -14).
   • Confirm using vertex formula: -b/2a
     • For b = 6, a = 1: -6/2 = -3, confirming the x-coordinate of the vertex.

Converting Vertex Form to Standard Form

1. Expand Using FOIL Method

   • Given vertex form: (x + 3)^2 - 14
   • Expand (x + 3)^2 using FOIL:
     • x*x = x^2
     • x*3 = 3x
     • 3*x = 3x
     • 3*3 = 9
   • Combine like terms:
     • 3x + 3x = 6x
     • 9 - 14 = -5

2. Resulting Standard Form

   • The expanded form gives you the standard quadratic form: x^2 + 6x - 5

Key Takeaways

• Standard to Vertex Form: Complete the square to rewrite the quadratic as a perfect square trinomial.
• Vertex to Standard Form: Expand the squared term using the FOIL method to revert to standard form.
• Understanding these conversions helps in analyzing and graphing quadratic functions.