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Completing the square is a method used to convert a quadratic equation into its vertex form. This is helpful for easy graphing and understanding properties of the parabola.
To complete the square for the quadratic function:
1. Start with the equation: \( y = -3x^2 + 6x - 3 \).
2. Factor out the coefficient of \( x^2 \) from the first two terms: \( y = -3(x^2 - 2x) - 3 \).
3. Take half of the linear term's coefficient (\b) squared and add and subtract it inside the parenthesis: \( y = -3(x^2 - 2x + 1 - 1) - 3 \).
4. Simplify, group the perfect square inside the parenthesis, and distribute the constant: \( y = -3((x - 1)^2 - 1) - 3\).
5. Finally, combine constants to get the vertex form: \( y = -3(x - 1)^2 + 3 - 3 \), which simplifies to \( y = -3(x-1)^2 \).
Using this method, the equation is transformed into a form that easily reveals the vertex of the parabola.

The vertex form of a quadratic function is \(y = a(x-h)^2 + k\). This form is beneficial because it clearly shows the vertex \((h, k)\) and the direction the parabola opens.
The quadratic function we started with, \( y = -3x^2 + 6x - 3 \), has been converted to vertex form: \( y = -3(x-1)^2 \). This indicates:

• The vertex \( (h, k)\) is at \( (1, 0)\).
• The coefficient \( a = -3 \) means the parabola opens downwards and is vertically stretched by a factor of 3.

Using the vertex form allows us to quickly identify transformations from the base parabola \( y = x^2 \), such as reflections, translations, and dilations.

Graphing a parabola is straightforward once it is in vertex form: \( y = a(x-h)^2 + k \).
For the quadratic function \( y = -3(x-1)^2 \), follow these steps:

• Identify the vertex: here it is \( (1, 0) \).
• Determine the direction the parabola opens: since \( a = -3 \), it opens downward.
• Consider the vertical stretch factor: here, the stretch factor is 3, making the parabola narrower than \( y = x^2 \).
• Plot the vertex and additional points around it to visualize the shape.
• Use symmetry about the vertex to plot corresponding points on either side.

By following these steps, you can accurately sketch the graph of \( y = -3(x-1)^2 \). The process reveals the transformation applied to the basic \( y = x^2 \) parabola to obtain the desired shape.