Section 2.2: Vertex Form & Transformations

This section focuses on rewriting quadratics in vertex form and understanding transformations: shifts, stretches, compressions, and reflections.

Example 1: Convert to Vertex Form

Rewrite \( f(x) = x^2 - 6x + 8 \) in vertex form.

Step 1: Complete the square: \( x^2 - 6x + 8 = (x^2 - 6x + 9) - 1 = (x-3)^2 -1 \)

Step 2: Vertex form: \( f(x) = (x-3)^2 -1 \), Vertex: (3, -1)

Example 2: Transformations

Describe the transformations of \( g(x) = -2(x+1)^2 + 3 \).

  • Reflection over x-axis: negative coefficient
  • Vertical stretch by factor 2
  • Horizontal shift left 1 unit
  • Vertical shift up 3 units

Practice Problems

  1. Rewrite \( f(x) = x^2 + 8x + 12 \) in vertex form
  2. Rewrite \( h(x) = -x^2 + 4x - 1 \) in vertex form
  3. Describe the transformations of \( k(x) = 3(x-2)^2 -5 \)
  4. Find the vertex of \( m(x) = -2x^2 - 8x + 6 \)
  5. Graph \( f(x) = (x+3)^2 - 4 \) and identify shifts
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