Vertex Form of Quadratic Equation

Introduction

• Different textbooks refer to the "standard form" of a quadratic function differently:
  • Some use: ( f(x) = ax^2 + bx + c )
  • Others use: ( f(x) = a(x - h)^2 + k )
• To avoid confusion, this guide uses "vertex form" for ( f(x) = a(x - h)^2 + k ) and refers to ( f(x) = ax^2 + bx + c ) by its full statement.

Vertex Form

• Vertex form of a quadratic function: ( f(x) = a(x - h)^2 + k )
• ((h, k)) is the vertex of the parabola, the "turning point"
  • h represents a horizontal shift
  • k represents a vertical shift
• Example:
  • ( y = 2(x - 1)^2 + 5 ) implies ( h = 1, k = 5 )
  • ( y = 3(x + 4)^2 - 6 ) implies ( h = -4, k = -6 )

Converting from ( ax^2 + bx + c ) to Vertex Form

Method 1: Completing the Square

• Example: Convert ( y = 2x^2 - 4x + 5 ) to vertex form
  • Isolate ( x^2 ) and ( x ) terms:
    • ( y - 5 = 2x^2 - 4x )
  • Factor the leading coefficient of 2:
    • ( y - 5 = 2(x^2 - 2x) )
  • Create a perfect square trinomial:
    • Half the coefficient of ( x ), square it, and add it inside the parentheses
  • Simplify and solve:
    • ( y - 3 = 2(x - 1)^2 )
    • ( y = 2(x - 1)^2 + 3 )
  • Vertex: (1, 3)

Method 2: Using the "Sneaky Tidbit"

• Example equation: ( y = 2x^2 - 4x + 5 )
• Vertex ((h, k)): Use calculations to find ((h, k) = (1, 3))
• Write the vertex form: ( y = 2(x - 1)^2 + 3 )

Converting from Vertex Form to ( ax^2 + bx + c )

• Multiply out and combine like terms:
  • ( y = 2(x - 1)^2 + 3 )
  • Expand and simplify to obtain ( y = 2x^2 - 4x + 5 )

Graphing a Quadratic in Vertex Form

1. Start with the function in vertex form: ( y = a(x - h)^2 + k )
2. Identify and plot the vertex ((h, k))
3. Draw the axis of symmetry ( x = h )
4. Find points by substituting ( x )-values:
   • Example points: (1, -1) and (0, 8)
5. Plot mirror points across the axis of symmetry
6. Draw the parabola (ensure it is curved, not straight)

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