Summary
Highlights
This video explains how to convert a quadratic equation from standard form (y = ax^2 + bx + c) to vertex form (y = a(x - h)^2 + k). This conversion is useful for graphing and easily identifying the vertex of a parabola. The method involves creating a perfect square trinomial through completing the square.
The first step is to ensure that the coefficient 'a' of the x^2 term is 1. If 'a' is not 1, factor it out from the first two terms (ax^2 + bx). Examples are provided: for y = x^2 + 4x - 5, 'a' is already 1; for y = -x^2 - 6x + 11, factor out -1; for y = 5x^2 + 10x + 1, factor out 5.
Focus on the quadratic and linear terms (x^2 and x terms). To create a perfect square trinomial, calculate (b/2)^2. For the first example (x^2 + 4x), (4/2)^2 = 4. For the second example (x^2 - 6x), (-6/2)^2 = 9. For the third example (x^2 + 2x), (2/2)^2 = 1. This value will be added inside the parentheses.
Add the calculated value inside the parentheses to create the perfect square trinomial. To maintain the equality of the equation, subtract the 'adjusted' value outside the parentheses. The adjustment means multiplying the added value by the 'a' that was factored out earlier. For example, if a -1 was factored out and 9 was added inside, then -1 * 9 = -9 should be subtracted outside.
A perfect square trinomial factors into a binomial squared. For x^2 + 4x + 4, it factors to (x + 2)^2. Combine the constant terms outside the parentheses. This results in the equation being in vertex form. Three examples are walked through to demonstrate the final conversion to y = a(x - h)^2 + k.