Finding The Vertex Coordinates Of Quadratic Function F(x) = X² + 10x - 3

by ADMIN 73 views

In the realm of mathematics, quadratic functions hold a prominent position, often encountered in various applications ranging from physics to engineering. Understanding the properties of these functions is crucial for solving real-world problems. One such property is the vertex, a critical point that reveals the function's minimum or maximum value. In this comprehensive guide, we will delve into the process of determining the coordinates of the vertex for the quadratic function f(x) = x² + 10x - 3. We will explore different methods, providing a clear understanding of the underlying concepts and techniques.

Understanding Quadratic Functions and the Vertex

Before we embark on the journey of finding the vertex, let's first establish a solid understanding of quadratic functions and the significance of the 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 ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient a.

The vertex of a parabola is the point where the function reaches its minimum or maximum value. If a > 0, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if a < 0, the parabola opens downwards, and the vertex represents the maximum point of the function. The vertex plays a crucial role in understanding the behavior of the quadratic function and its applications.

Methods for Finding the Vertex

There are several methods to determine the coordinates of the vertex of a quadratic function. We will explore three common techniques:

  1. Using the Vertex Formula: The vertex formula provides a direct approach to calculate the coordinates of the vertex. For a quadratic function in the form f(x) = ax² + bx + c, the vertex is located at the point (-b/2a, f(-b/2a)). This formula is derived from the process of completing the square and provides a straightforward way to find the vertex coordinates.
  2. Completing the Square: Completing the square is a technique used to rewrite a quadratic expression in a form that reveals the vertex. By manipulating the equation f(x) = ax² + bx + c, we can transform it into the form f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. This method provides a deeper understanding of the relationship between the quadratic expression and its vertex.
  3. Using Calculus (Finding the Critical Point): Calculus provides a powerful tool for finding the vertex of a quadratic function. The vertex corresponds to the critical point of the function, where the derivative equals zero. By finding the derivative of the function f(x) = ax² + bx + c and setting it equal to zero, we can solve for the x-coordinate of the vertex. The y-coordinate can then be found by substituting the x-coordinate back into the original function. This method demonstrates the connection between calculus and the properties of quadratic functions.

Applying the Vertex Formula to f(x) = x² + 10x - 3

Let's now apply the vertex formula to find the coordinates of the vertex for the function f(x) = x² + 10x - 3. Comparing this function with the standard form f(x) = ax² + bx + c, we identify the coefficients as a = 1, b = 10, and c = -3.

Using the vertex formula, the x-coordinate of the vertex is given by:

x = -b / 2a = -10 / (2 * 1) = -5

To find the y-coordinate, we substitute the x-coordinate back into the function:

f(-5) = (-5)² + 10(-5) - 3 = 25 - 50 - 3 = -28

Therefore, the coordinates of the vertex for the function f(x) = x² + 10x - 3 are (-5, -28).

Completing the Square for f(x) = x² + 10x - 3

Now, let's use the method of completing the square to verify our result. Starting with the function f(x) = x² + 10x - 3, we aim to rewrite it in the form f(x) = a(x - h)² + k.

  1. Focus on the x² and x terms: x² + 10x
  2. Take half of the coefficient of the x term (which is 10), square it (which is 25), and add and subtract it within the expression: x² + 10x + 25 - 25 - 3
  3. Rewrite the first three terms as a squared expression: (x + 5)² - 25 - 3
  4. Simplify the constant terms: (x + 5)² - 28

Now, the function is in the form f(x) = (x + 5)² - 28, which can be rewritten as f(x) = 1(x - (-5))² + (-28). Comparing this with the form f(x) = a(x - h)² + k, we see that the vertex is indeed at (-5, -28).

Using Calculus to Find the Vertex of f(x) = x² + 10x - 3

Finally, let's employ calculus to determine the vertex. The derivative of the function f(x) = x² + 10x - 3 is:

f'(x) = 2x + 10

To find the critical point, we set the derivative equal to zero and solve for x:

2x + 10 = 0 2x = -10 x = -5

This confirms that the x-coordinate of the vertex is -5. Substituting this value back into the original function, we get:

f(-5) = (-5)² + 10(-5) - 3 = -28

Again, we arrive at the vertex coordinates (-5, -28).

Conclusion

In this comprehensive guide, we have explored the concept of the vertex of a quadratic function and demonstrated various methods to find its coordinates. We applied the vertex formula, completed the square, and utilized calculus to determine the vertex of the function f(x) = x² + 10x - 3. Through these methods, we consistently found the vertex to be located at (-5, -28). Understanding the vertex and its significance is crucial for analyzing and applying quadratic functions in diverse fields. This knowledge empowers us to solve optimization problems, model real-world phenomena, and gain a deeper appreciation for the elegance and power of mathematics. The correct answer is A. (-5, -28). This exploration not only provides the solution to the given problem but also equips readers with a comprehensive understanding of quadratic functions and vertex determination techniques.