Transforming Quadratic Functions from General Form to Vertex Form

Transforming a quadratic function from General Form, ( y ax^2 bx c ), to Vertex Form, ( y a(x - h)^2 k ), can be a powerful technique in understanding and analyzing quadratic equations. This process, known as completing the square, is crucial in many areas of mathematics, including graphing, solving quadratic equations, and finding the vertex of a parabola. In this article, we'll walk through the steps to transform the given quadratic function using the process of completing the square.

Step-by-Step Guide: Completing the Square

Consider the given quadratic function:

Given Function: ( y x^2 - 2x 3 )

Step 1: Factor out the Coefficient of ( x^2 )

In this case, the coefficient of ( x^2 ) is 1. Since it is 1, we can skip this step. If it were a different value, we would factor it out.

Step 2: Complete the Square

The next step is to complete the square. Here’s how to do it:

Take the coefficient of ( x ), which is (-2). Divide it by 2 and square the result. Substitute it back into the equation. Adjust the equation to maintain the equality.

Let’s go through the process:

Take the coefficient of ( x ): (-2). Divide it by 2: (frac{-2}{2} -1). Square the result: ((-1)^2 1). Add and subtract this square inside the equation: ( y x^2 - 2x 1 - 1 3 ) Rearrange the equation: ( y (x^2 - 2x 1) (3 - 1) ) Which simplifies to: ( y (x - 1)^2 2 )

Now, the function is in the vertex form ( y a(x - h)^2 k ).

Step 3: Write in Vertex Form

The vertex form of the quadratic function ( y x^2 - 2x 3 ) is:

( y (x - 1)^2 2 )

In this form, the vertex ( (h, k) ) is ( (1, 2) ).

Summary

The vertex form of the given quadratic function ( y x^2 - 2x 3 ) is ( y (x - 1)^2 2 ).

By following these steps, you can transform any quadratic function from general form to vertex form. This transformation is particularly useful in analyzing the graph of the quadratic function, as the vertex form directly reveals the vertex of the parabola.

Practical Examples and Tips

Here are a few more examples to solidify your understanding:

Example 1

Transform the function ( y 2x^2 - 8x - 5 ).

Factor out the coefficient of ( x^2 ): ( y 2(x^2 - 4x) - 5 ). Complete the square inside the parentheses: Take the coefficient of ( x ): (-4). Divide it by 2: (frac{-4}{2} -2). Square the result: ((-2)^2 4). Add and subtract this square inside the parentheses: ( y 2(x^2 - 4x 4 - 4) - 5 ) Rearrange the equation: ( y 2((x - 2)^2 - 4) - 5 ) Simplify further: ( y 2(x - 2)^2 - 8 - 5 ) ( y 2(x - 2)^2 - 13 )

The vertex form is ( y 2(x - 2)^2 - 13 ).

Example 2

Transform the function ( y 3x^2 6x 2 ).

Factor out the coefficient of ( x^2 ): ( y 3(x^2 2x) 2 ). Complete the square inside the parentheses: Take the coefficient of ( x ): (2). Divide it by 2: (frac{2}{2} 1). Square the result: (1^2 1). Add and subtract this square inside the parentheses: ( y 3(x^2 2x 1 - 1) 2 ) Rearrange the equation: ( y 3((x 1)^2 - 1) 2 ) Simplify further: ( y 3(x 1)^2 - 3 2 ) ( y 3(x 1)^2 - 1 )

The vertex form is ( y 3(x 1)^2 - 1 ).

Remember, the key steps are to identify the coefficient of ( x ), divide it by 2, square the result, and then adjust the equation to maintain the equality.

Conclusion

Transforming quadratic functions from general form to vertex form is a valuable skill in algebra and is often used in graphing and equation solving. By understanding and mastering this technique, you can efficiently convert quadratic functions and derive useful information about their properties. Practice these steps with various functions to become proficient.