Which equation is $y = -3x^2 - 12x - 2$ rewritten in vertex form?
We are given the quadratic equation in standard form: \[y = -3x^2 + 12x + 2\] Our objective is to rewrite this equation in the vertex form, which is given by \[y = a(x - h)^2 + k\] where \((h, k)\) represents the coordinates of the vertex of the parabola. Converting from standard form to vertex form allows us to easily identify the vertex and the direction of opening of the parabola.
To transform the equation \(y = -3x^2 + 12x + 2\) into vertex form, we use the method of completing the square. First, we factor out the coefficient of the \(x^2\) term, which is \(-3\), from the first two terms of the expression: \[y = -3(x^2 - 4x) + 2\] Next, we identify the value needed to complete the square for the expression inside the parentheses. We take half of the coefficient of \(x\), which is \(-4\), and square it: \(\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4\). To maintain the equality of the expression, we add and subtract \(4\) inside the parentheses: \[y = -3(x^2 - 4x + 4 - 4) + 2\] Now, we distribute the \(-3\) to the \(-4\) term and remove it from the parentheses: \[y = -3(x^2 - 4x + 4) + (-3)(-4) + 2\] \[y = -3(x^2 - 4x + 4) + 12 + 2\] Simplifying the constants gives: \[y = -3(x - 2)^2 + 14\] Thus, the vertex form of the equation is \(y = -3(x - 2)^2 + 14\). Comparing this result with the given options, we identify it as the second option.
Vertex form is extremely useful in real-world scenarios involving parabolic paths, such as the trajectory of a ball kicked in the air or water spraying from a fountain. By writing the quadratic equation in vertex form \(y = a(x - h)^2 + k\), we immediately know the maximum or minimum height (\(k\)) reached by the object and the exact horizontal distance (\(h\)) at which this occurs. This simplifies planning and analysis in fields like sports science, engineering, and architecture.