Problem 81 - Elementary Algebra

Question

Rewrite in \(y=a(x-h)_{2}+k\) form and determine the vertex. $$ y=x 2+4 x-12 $$

Step-by-Step Solution

Answer

The vertex form is \( y = (x + 2)^2 - 16 \) with vertex \((-2, -16)\).

Step 1: Rearrange the Equation

The given equation is \( y = x^2 + 4x - 12 \). The goal is to rewrite it in vertex form, \( y = a(x-h)^2 + k \). Start by identifying the quadratic and linear terms: \( x^2 \) and \( 4x \).

Step 2: Complete the Square

To complete the square, take the linear coefficient (which is 4), halve it to get 2, and square it to add 4. Thus, you adjust the equation to complete the square: \( y = (x^2 + 4x + 4) - 4 - 12 \). This maintains the equation's integrity by adding and subsequently subtracting 4.

Step 3: Factor the Perfect Square

Notice that \( x^2 + 4x + 4 \) is a perfect square. You can factor it as \( (x + 2)^2 \). Thus, rewrite the equation: \( y = (x + 2)^2 - 16 \).

Step 4: Write Equation in Vertex Form

The equation \( y = (x + 2)^2 - 16 \) is now in vertex form, \( y = a(x-h)^2 + k \), where \( a = 1 \), \( h = -2 \), and \( k = -16 \).

Step 5: Identify the Vertex

In the vertex form \( y = a(x-h)^2 + k \), the vertex \( (h, k) \) is \( (-h, k) \), so the vertex here is \( (-2, -16) \).

Key Concepts

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a form that makes identifying certain properties, like the vertex, easier. This is particularly useful in vertex form equations. The aim is to transform a quadratic expression into a perfect square trinomial. Here's how it's typically done:

• Start with the quadratic expression: for example, consider terms like \(x^2\) and \(4x\) in the equation \(y = x^2 + 4x - 12\).
• Take the linear coefficient (in this case, the 4 in \(4x\)), halve it (making it 2), and then square it (resulting in 4). This helps in forming a perfect square trinomial.
• The expression \(x^2 + 4x\) becomes \(x^2 + 4x + 4\), a perfect square trinomial that can be rewritten as \((x+2)^2\).

This process portrays how the quadratic part of the equation is manipulated into a form where the square of a binomial is visible. Remember that whatever you add inside the equation, you must also subtract to ensure balance is maintained.

Quadratic Equations

A quadratic equation is any equation that takes the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. These equations are called 'quadratic' because the highest power of the variable \(x\) is 2.
Quadratic equations hold a distinctive U-shaped graph called a parabola that opens upward if \(a > 0\) and downward if \(a < 0\). The general strategies to solve quadratic equations include:

• Factoring the quadratic expression.
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
• Completing the square to rearrange the equation into vertex form \(y = a(x-h)^2 + k\).

In our original exercise, the specific equation \(y = x^2 + 4x - 12\) was rearranged using completion of the square, resulting in a format that allowed easy identification of the vertex.

Vertex Identification

After successfully completing the square, the quadratic equation can be expressed in vertex form, \(y = a(x-h)^2 + k\). The beauty of this form lies in how it simplifies the process of finding the vertex of the parabola associated with the equation.
The vertex form directly shows the vertex of the parabola as the point \((h, k)\). In our example problem, the equation \(y = (x + 2)^2 - 16\) is already in this form:

• The constant \(a = 1\) indicates the parabola opens upward.
• The transformations inside the equation reveal the vertex \((-h, k)\) as \((-2, -16)\).

Knowing how to express a quadratic equation in its vertex form allows for quick and accurate determination of the vertex, which is crucial for graphing and analyzing the behavior of quadratic functions.