Vertex Form: Complete the Square to Find the Vertex
Rewrite a quadratic in vertex form by completing the square and read the vertex, with two worked Algebra 2 examples.
Vertex form
- y = a(x − h)² + k
- Vertex = (h, k)
h shifts left or right, k shifts up or down.
Complete the square
- Take half of the x coefficient
- Square it, add and subtract it
- Factor the perfect square
For x² + bx: add (b/2)² and subtract it.
Example 1
Write y = x² − 6x + 5 in vertex form.
- Half of −6 is −3, and (−3)² = 9
- y = (x² − 6x + 9) − 9 + 5
- y = (x − 3)² − 4
Vertex form: y = (x − 3)² − 4, vertex (3, −4)
Example 2
Find the vertex of y = x² + 4x + 1.
- Half of 4 is 2, and 2² = 4
- y = (x² + 4x + 4) − 4 + 1
- y = (x + 2)² − 3
Vertex (−2, −3)
Common mistake
y = (x − 3)² − 4 has its vertex at (−3, −4)
The sign inside flips: (x − 3) means h = +3.
The number outside keeps its sign: k = −4.
If a ≠ 1, factor a out of the x terms first.
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