Finding The Vertex Form Of Y=-3x²-12x-2 A Step-by-Step Guide

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In the realm of quadratic equations, the vertex form stands out as a particularly insightful representation. It provides a clear view of the parabola's vertex, the point where it reaches its maximum or minimum value. This article delves into the process of converting the quadratic equation y = -3x² - 12x - 2 into its vertex form, offering a step-by-step guide and highlighting the significance of this transformation.

Understanding the Vertex Form

The vertex form of a quadratic equation is expressed as:

y = a(x - h)² + k

Where:

  • (h, k) represents the vertex of the parabola.
  • a determines the direction and stretch of the parabola.

Our goal is to rewrite the given equation, y = -3x² - 12x - 2, in this format. This involves a technique called "completing the square," which allows us to manipulate the equation algebraically while preserving its inherent meaning.

The Significance of Vertex Form

Why bother converting to vertex form? The answer lies in the wealth of information it readily reveals. The vertex, (h, k), tells us the parabola's highest or lowest point, a crucial piece of data in various applications, such as optimization problems. The value of a not only indicates whether the parabola opens upwards (a > 0) or downwards (a < 0) but also dictates its width. A larger absolute value of a signifies a narrower parabola, while a smaller value results in a wider one.

Step-by-Step Conversion: Completing the Square

Now, let's embark on the journey of transforming y = -3x² - 12x - 2 into vertex form. This involves a series of algebraic manipulations that systematically reshape the equation.

Step 1: Factor out the Leading Coefficient

The first step is to factor out the coefficient of the x² term, which is -3 in our case, from the first two terms of the equation:

y = -3(x² + 4x) - 2

This step isolates the x² and x terms, preparing them for the next stage of completing the square.

Step 2: Complete the Square

Completing the square is the heart of this transformation. We aim to create a perfect square trinomial within the parentheses. To do this, we take half of the coefficient of the x term (which is 4), square it (4/2 = 2, 2² = 4), and add it inside the parentheses. However, to maintain the equation's balance, we must also subtract the same value multiplied by the factored-out coefficient (-3) outside the parentheses:

y = -3(x² + 4x + 4) - 2 + 12

Notice that we added 4 inside the parentheses but effectively subtracted -3 * 4 = -12 from the right side of the equation, which is why we add 12 outside the parentheses. The expression inside the parentheses is now a perfect square trinomial.

Step 3: Express as a Squared Term

The perfect square trinomial can be rewritten as a squared term:

y = -3(x + 2)² - 2 + 12

The expression (x² + 4x + 4) is equivalent to (x + 2)². This simplification is a direct result of completing the square.

Step 4: Simplify the Constant Term

Finally, we combine the constant terms outside the parentheses:

y = -3(x + 2)² + 10

This is the vertex form of the equation. We have successfully rewritten the original equation in the desired format.

Identifying the Vertex and Analyzing the Parabola

Now that we have the equation in vertex form, y = -3(x + 2)² + 10, we can readily identify the vertex and analyze the parabola's characteristics.

The Vertex

Comparing our equation to the general vertex form, y = a(x - h)² + k, we can see that:

  • h = -2
  • k = 10

Therefore, the vertex of the parabola is at the point (-2, 10). This is the highest point on the parabola since the coefficient a is negative.

Direction and Stretch

The coefficient a is -3. Since it's negative, the parabola opens downwards. The absolute value of a is 3, which indicates that the parabola is narrower than the standard parabola y = x².

Practical Applications of Vertex Form

The vertex form isn't just a mathematical curiosity; it has practical applications in various fields.

Optimization Problems

In optimization problems, we often seek to find the maximum or minimum value of a function. When dealing with quadratic functions, the vertex directly provides this information. For instance, if we were modeling the trajectory of a projectile, the vertex would represent the projectile's maximum height.

Graphing Quadratic Functions

Vertex form makes graphing quadratic functions significantly easier. We can plot the vertex and then use the value of a to determine the parabola's shape and direction. This allows for a quick and accurate sketch of the graph.

Real-World Scenarios

Quadratic functions model various real-world phenomena, such as the shape of suspension bridges, the path of a ball thrown in the air, and the design of parabolic reflectors. Understanding vertex form helps us analyze and interpret these scenarios.

Conclusion

Converting a quadratic equation to vertex form is a powerful technique that unlocks valuable insights into the parabola's behavior. By completing the square, we can rewrite the equation in a form that explicitly reveals the vertex, direction, and stretch of the parabola. This knowledge is crucial for solving optimization problems, graphing quadratic functions, and understanding real-world applications modeled by quadratic equations. The equation y = -3x² - 12x - 2 when rewritten in vertex form becomes y = -3(x + 2)² + 10, a testament to the elegance and utility of this mathematical transformation.