Understanding Vertex Form

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form clearly shows the transformations applied to the basic parabola y = x²:

a determines vertical stretch/compression and direction
h determines horizontal translation
k determines vertical translation

Conversion Steps

Start with standard form: y = ax² + bx + c
Complete the square to group x terms
Factor and simplify to vertex form
Identify vertex (h, k) from the resulting equation

Example Conversion

For y = 2x² - 8x + 6:

Factor out a: y = 2(x² - 4x) + 6
Complete the square: y = 2(x² - 4x + 4) + 6 - 8
Simplify: y = 2(x - 2)² - 2
Vertex: (2, -2)

Frequently Asked Questions

What is the advantage of vertex form?

Vertex form immediately reveals the vertex coordinates and the parabola's direction, making it easier to graph and analyze transformations.

Can all quadratics be converted to vertex form?

Yes, all quadratic equations can be written in vertex form through the process of completing the square.

How does the 'a' coefficient affect the parabola?

The |a| value determines how "steep" the parabola is. If a > 0, it opens upward; if a < 0, it opens downward.

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