Find The Quadratic Function With Vertex At (2, -9)

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In mathematics, identifying a quadratic function with a specific vertex is a common problem. This article delves into the process of determining which quadratic function, among a given set of options, possesses a vertex at the point (2, -9). We will explore the properties of quadratic functions, their vertex form, and how to manipulate equations to identify the correct function. This comprehensive guide aims to provide a clear understanding of the concepts and techniques involved. Understanding quadratic functions and their graphical representations is crucial in various fields, including physics, engineering, and economics. For example, the trajectory of a projectile, the shape of a suspension bridge cable, and the optimization of business costs can all be modeled using quadratic functions. Therefore, mastering the ability to identify key features of quadratic functions, such as the vertex, is essential for problem-solving in these diverse areas.

Understanding Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the standard form as 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. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The vertex of the parabola is the point where the curve changes direction; it is the minimum point if the parabola opens upwards and the maximum point if the parabola opens downwards. The vertex form of a quadratic function provides a direct way to identify the vertex. This form is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. Converting a quadratic function from standard form to vertex form involves completing the square, a technique that rewrites the quadratic expression as a perfect square plus a constant. This process is fundamental in identifying the vertex and understanding the parabola's transformations from the basic y = x² parabola. By analyzing the vertex form, we can easily determine the vertex (h, k), the axis of symmetry (x = h), and whether the parabola opens upwards or downwards based on the sign of a. Understanding these properties is crucial for sketching the graph of a quadratic function and solving related problems.

The Significance of the Vertex

The vertex of a parabola is a critical point that reveals essential information about the quadratic function. As mentioned earlier, the vertex represents the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards. In practical applications, the vertex can signify the optimal point in a system. For instance, in a business context, the vertex might represent the production level that minimizes cost or maximizes profit. In physics, the vertex could indicate the highest point reached by a projectile or the lowest point of a hanging cable. The x-coordinate of the vertex also gives the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. This symmetry is a fundamental property of parabolas and simplifies graphing and analysis. Moreover, the vertex form of a quadratic function, f(x) = a(x - h)² + k, directly incorporates the vertex coordinates (h, k), making it easy to identify the vertex without further calculations. The value of k in the vertex form represents the y-coordinate of the vertex, which is the minimum or maximum value of the function. The value of h represents the x-coordinate of the vertex, which is the axis of symmetry. Therefore, understanding the significance of the vertex and the vertex form of a quadratic function is crucial for solving various mathematical problems and real-world applications. The ability to quickly identify the vertex allows for efficient analysis and optimization in numerous scenarios.

Identifying the Function with Vertex (2, -9)

To find the quadratic function with a vertex at (2, -9), we need to analyze the given options and see which one can be transformed into the vertex form f(x) = a(x - h)² + k with h = 2 and k = -9. This means we are looking for a function that can be written in the form f(x) = a(x - 2)² - 9, where a is a constant. The given options are:

  1. f(x) = -(x - 3)²
  2. f(x) = (x + 8)²
  3. f(x) = (x - 5)(x + 1)
  4. f(x) = -(x - 1)(x - 5)

Let's examine each option:

Option 1: f(x) = -(x - 3)²

This function is already in vertex form, but the vertex is at (3, 0), not (2, -9). Therefore, this option is incorrect. The vertex form of this function is f(x) = -1(x - 3)² + 0, which clearly shows the vertex at (3, 0). The coefficient a is -1, indicating that the parabola opens downwards, but the vertex coordinates do not match our target (2, -9).

Option 2: f(x) = (x + 8)²

This function is also in vertex form, but the vertex is at (-8, 0), not (2, -9). Thus, this option is also incorrect. The vertex form of this function is f(x) = 1(x + 8)² + 0, which indicates the vertex at (-8, 0). The coefficient a is 1, meaning the parabola opens upwards, but the vertex does not match our required coordinates.

Option 3: f(x) = (x - 5)(x + 1)

This function is in factored form. To determine its vertex, we first need to expand it into standard form and then convert it to vertex form. Expanding the function, we get:

f(x) = x² + x - 5x - 5 f(x) = x² - 4x - 5

Now, we complete the square to convert it to vertex form:

f(x) = (x² - 4x) - 5 f(x) = (x² - 4x + 4) - 5 - 4 f(x) = (x - 2)² - 9

This function is now in vertex form, f(x) = (x - 2)² - 9, and the vertex is at (2, -9). Therefore, this is the correct function.

Option 4: f(x) = -(x - 1)(x - 5)

This function is also in factored form. Expanding the function, we get:

f(x) = -(x² - 5x - x + 5) f(x) = -(x² - 6x + 5) f(x) = -x² + 6x - 5

Now, we complete the square to convert it to vertex form:

f(x) = -(x² - 6x) - 5 f(x) = -(x² - 6x + 9) - 5 + 9 f(x) = -(x - 3)² + 4

This function is in vertex form, f(x) = -(x - 3)² + 4, and the vertex is at (3, 4), not (2, -9). Therefore, this option is incorrect.

Step-by-Step Solution

Let's recap the process we used to identify the function with the vertex at (2, -9):

  1. Understand the Vertex Form: Recognize that the vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex.
  2. Analyze the Given Options: Examine each function provided and determine if it is already in vertex form or factored form.
  3. Convert to Vertex Form: If the function is in factored form, expand it into standard form (f(x) = ax² + bx + c) and then complete the square to convert it to vertex form.
  4. Identify the Vertex: Once in vertex form, identify the vertex (h, k) by comparing the function to the general vertex form equation.
  5. Match the Vertex: Compare the vertex of each function to the target vertex (2, -9) and select the function that matches.

In our case, we found that option 3, f(x) = (x - 5)(x + 1), when expanded and converted to vertex form, becomes f(x) = (x - 2)² - 9, which has a vertex at (2, -9).

Conclusion

Identifying the quadratic function with a specific vertex involves understanding the properties of quadratic functions, particularly the vertex form. By converting functions to vertex form and comparing the vertex coordinates, we can accurately determine the function that meets the given criteria. In this case, the function f(x) = (x - 5)(x + 1) has a vertex at (2, -9). Mastering these techniques is essential for solving a wide range of mathematical problems and real-world applications involving quadratic functions. The vertex plays a crucial role in understanding the behavior and characteristics of a parabola, making it a key element in quadratic function analysis. The process of converting between standard form, factored form, and vertex form provides a comprehensive understanding of quadratic functions and their graphical representations. This knowledge is invaluable for students and professionals alike in various fields where quadratic functions are applied.