Kellen's error is in the last step where he expands the expression (x+3)² incorrectly. The correct vertex form of the given function is f(x) = (x + 3)² - 3.
Kellen's error is in the last step where he expands the expression (x+3)². The correct expansion would be (x+3)(x+3), which simplifies to x²+6x+9. However, Kellen incorrectly expanded it to x²+6, which led to the incorrect vertex form of the equation.
To convert the function f(x) = x² + 6x + 6 to vertex form, we need to complete the square. The vertex form of a quadratic function is f(x) = a(x-h)² + k, where (h, k) is the vertex of the parabola.
First, factor out the leading coefficient 1 from the terms with x: f(x) = 1(x² + 6x) + 6.
Next, complete the square by adding and subtracting half of the coefficient of x, squared: f(x) = 1(x² + 6x + 9 - 9) + 6.
Factor the trinomial x² + 6x + 9 as a perfect square: f(x) = 1((x + 3)² - 9) + 6.
Simplify the expression inside the parentheses: f(x) = (x + 3)² - 9 + 6.
Combine like terms: f(x) = (x + 3)² - 3.
Therefore, the correct vertex form of the given function is f(x) = (x + 3)² - 3.