Transforming Quadratic Functions: Expressing ( f(x) 2x^2 - 4x - 13 ) in Vertex Form

Quadratic functions are often represented in standard form, but it is useful to express them in vertex form for various applications. In this article, we will explore how to manipulate the quadratic function ( f(x) 2x^2 - 4x - 13 ) to express it in the form ( f(x) a(x - h)^2 k ). By doing so, we reveal the vertex of the parabola, which provides valuable information about the maximum or minimum point of the function.

Understanding the Vertex Form

The vertex form of a quadratic function is ( f(x) a(x - h)^2 k ), where the vertex of the parabola is given by the point ( (h, k) ). This form is especially useful for graphing and understanding the behavior of the function. To convert the given function into vertex form, we will complete the square. Let's break down the process step-by-step.

Steps to Complete the Square

Step 1: Factor Out the Leading Coefficient

First, we factor out the leading coefficient from the ( x^2 ) and ( x ) terms. In our case, the leading coefficient is 2.

$$ f(x) 2x^2 - 4x - 13 f(x) 2(x^2 - 2x) - 13 $$

Step 2: Complete the Square

To complete the square, we need to form a perfect square trinomial inside the parentheses. We do this by taking half of the coefficient of ( x ), squaring it, and adding and subtracting this value inside the parentheses.

The coefficient of ( x ) is -2. Half of -2 is -1. Squaring -1 gives 1. $$ f(x) 2(x^2 - 2x 1 - 1) - 13 f(x) 2((x - 1)^2 - 1) - 13 $$

Step 3: Distribute the Coefficient of ( x^2 )

Multiply 2 into the terms inside the parentheses and simplify the expression.

$$ f(x) 2(x - 1)^2 - 2 - 13 f(x) 2(x - 1)^2 - 15 $$

Identifying ( a ), ( h ), and ( k )

Now the function is in the desired form ( f(x) a(x - h)^2 k ), where:

( a 2 ) ( h 1 ) ( k -15 )

Thus, the function ( f(x) 2x^2 - 4x - 13 ) can be expressed as:

$$ f(x) 2(x - 1)^2 - 15 $$

Why is Completing the Square Important?

Completing the square is not just a technique for converting quadratic functions. It pops up in various algebraic contexts and is particularly useful in:

1. Finding the Vertex of a Parabola

The vertex form directly provides the vertex of the parabola, which is crucial for understanding the minimum or maximum value of the function.

2. Graphing Quadratic Functions

The vertex form makes it easier to graph the parabola because it clearly shows the vertex and the direction of the parabola (upward or downward).

3. Solving Quadratic Equations

Completing the square can also be used to solve quadratic equations, especially those that are not easily factored.

Conclusion

In conclusion, expressing a quadratic function in the form ( f(x) a(x - h)^2 k ) through completing the square is a powerful technique with various practical applications. By converting the given function ( f(x) 2x^2 - 4x - 13 ) into ( f(x) 2(x - 1)^2 - 15 ), we have highlighted its vertex at ( (1, -15) ), which is a critical point for understanding its behavior.

Whether you're a student, a teacher, or anyone involved in higher mathematics, mastering the art of completing the square can significantly enhance your problem-solving skills in algebra.