Convert y = ax² + bx + c into the vertex form y = a(x - p)² + q. This operation is called completing the square.
Example ①
Convert y = x² - 4x + 7 into vertex form and find its vertex.
Answer: y = (x² - 4x + 4) - 4 + 7 = (x - 2)² + 3. Vertex (2, 3)
Convert y = x² - 4x + 7 into vertex form and find its vertex.
Answer: y = (x² - 4x + 4) - 4 + 7 = (x - 2)² + 3. Vertex (2, 3)
Example ②
Convert y = 2x² + 8x + 5 into vertex form.
Answer: y = 2(x² + 4x) + 5 = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3. Vertex (-2, -3)
Convert y = 2x² + 8x + 5 into vertex form.
Answer: y = 2(x² + 4x) + 5 = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3. Vertex (-2, -3)
Key Point
- When a ≠ 1, factor a out of the x² and x terms before completing the square
- The x-coordinate of the vertex can also be found using x = -b/2a
Practice Problems
- Convert y = x² + 6x + 4 into vertex form and find its vertex.
- Find the vertex of y = -2x² + 4x - 1.
Answers and Explanations
- Answer: y = (x+3)² - 5, vertex (-3, -5).
- Answer: Vertex (1, 1). -2(x-1)² + 2 - 1 = -2(x-1)² + 1.