Finding The Vertex Of F(x) = X² + 12x A Step-by-Step Guide

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Finding the vertex of a quadratic function is a fundamental concept in algebra, offering valuable insights into the behavior and characteristics of the parabola it represents. In this comprehensive guide, we will delve into the intricacies of determining the vertex, specifically focusing on the function f(x) = x² + 12x. We will explore different methods, provide step-by-step explanations, and address common misconceptions, ensuring a thorough understanding of this essential topic.

Understanding Quadratic Functions and Their Vertex

Before diving into the specifics of our function, let's establish a solid foundation by understanding quadratic functions and their vertex. A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). The vertex of the parabola is the point where the curve changes direction, representing either the minimum or maximum value of the function.

The vertex plays a crucial role in analyzing the quadratic function. It helps us determine the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The x-coordinate of the vertex represents the axis of symmetry. Furthermore, the vertex helps us identify the range of the function, which is the set of all possible output values. If the parabola opens upwards, the y-coordinate of the vertex represents the minimum value of the function. Conversely, if the parabola opens downwards, the y-coordinate represents the maximum value.

Methods for Finding the Vertex

There are several methods for finding the vertex of a quadratic function, each with its own advantages and applications. We will explore two primary methods: completing the square and using the vertex formula.

1. Completing the Square

Completing the square is a powerful algebraic technique that transforms a quadratic expression into a perfect square trinomial, allowing us to easily identify the vertex. Let's apply this method to our function, f(x) = x² + 12x.

The process involves manipulating the equation to create a perfect square trinomial, which is an expression that can be factored into the form (x + h)² or (x - h)². To complete the square, we follow these steps:

  1. Focus on the x² and x terms: In our function, these terms are x² and 12x.
  2. Take half of the coefficient of the x term: The coefficient of the x term is 12, so half of it is 6.
  3. Square the result: Squaring 6 gives us 36.
  4. Add and subtract this value inside the equation: We add and subtract 36 within the equation to maintain its balance. This gives us: f(x) = x² + 12x + 36 - 36
  5. Factor the perfect square trinomial: The first three terms, x² + 12x + 36, form a perfect square trinomial that can be factored as (x + 6)². f(x) = (x + 6)² - 36

Now, the equation is in vertex form, which is f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Comparing our equation to the vertex form, we can identify the vertex as (-6, -36). The x-coordinate is the opposite of the value inside the parenthesis, and the y-coordinate is the constant term.

2. Using the Vertex Formula

The vertex formula provides a direct way to calculate the coordinates of the vertex without completing the square. For a quadratic function in the form f(x) = ax² + bx + c, the vertex formula is given by:

  • x-coordinate of vertex (h) = -b / 2a
  • y-coordinate of vertex (k) = f(h)

Let's apply the vertex formula to our function, f(x) = x² + 12x.

  1. Identify the coefficients: In our function, a = 1, b = 12, and c = 0.
  2. Calculate the x-coordinate: Using the formula, h = -b / 2a = -12 / (2 * 1) = -6.
  3. Calculate the y-coordinate: Substitute the x-coordinate (h = -6) into the function to find the y-coordinate: k = f(-6) = (-6)² + 12(-6) = 36 - 72 = -36

Therefore, the vertex of the function is (-6, -36), which confirms our result obtained through completing the square.

Analyzing the Vertex and Its Significance

Having determined the vertex of the function f(x) = x² + 12x to be (-6, -36), let's analyze its significance in the context of the parabola.

  • Axis of Symmetry: The x-coordinate of the vertex, -6, represents the axis of symmetry. This means that the parabola is symmetrical about the vertical line x = -6.
  • Minimum Value: Since the coefficient of the x² term (a = 1) is positive, the parabola opens upwards. Therefore, the vertex represents the minimum point of the function. The minimum value of the function is the y-coordinate of the vertex, which is -36.
  • Range: The range of the function is the set of all possible output values. Since the parabola opens upwards and the minimum value is -36, the range is [-36, ∞), meaning all y-values greater than or equal to -36.

Common Mistakes and Misconceptions

When working with quadratic functions and their vertices, certain mistakes and misconceptions can arise. Let's address some of the common ones:

  • Incorrect Sign in Vertex Formula: A common mistake is to use the wrong sign in the vertex formula. Remember that the x-coordinate of the vertex is given by -b / 2a, not b / 2a. Paying close attention to the negative sign is crucial.
  • Confusing Vertex Form: When completing the square, it's essential to correctly identify the vertex from the vertex form f(x) = a(x - h)² + k. Remember that the x-coordinate of the vertex is the opposite of the value inside the parenthesis (h), and the y-coordinate is the constant term (k).
  • Misinterpreting the Vertex: The vertex represents either the minimum or maximum value of the function, depending on the direction the parabola opens. If the parabola opens upwards (a > 0), the vertex is the minimum point. If the parabola opens downwards (a < 0), the vertex is the maximum point.

Conclusion

Finding the vertex of a quadratic function is a fundamental skill in algebra, providing valuable insights into the function's behavior and characteristics. In this guide, we have explored the function f(x) = x² + 12x in detail, demonstrating two primary methods for determining the vertex: completing the square and using the vertex formula. We have also discussed the significance of the vertex, its role in identifying the axis of symmetry, minimum/maximum values, and the range of the function. By understanding these concepts and avoiding common mistakes, you can confidently analyze and interpret quadratic functions and their parabolas.

Therefore, the correct answer to the question "What is the vertex of the function f(x) = x² + 12x?" is (-6, -36).