Range Of The Quadratic Function F(x) = -(x+5)(x+1) Explained

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When delving into the world of functions, a crucial aspect to grasp is the range. The range of a function encompasses all possible output values (y-values) that the function can produce. For a quadratic function, which is a polynomial function of degree two, the graph takes the shape of a parabola. This parabolic shape dictates the function's range, as it either opens upwards or downwards, leading to a minimum or maximum value, respectively. In this comprehensive exploration, we will dissect the quadratic function f(x) = -(x+5)(x+1) to determine its range. Understanding the range of quadratic functions is a fundamental concept in algebra and calculus. The range tells us the set of all possible output values of the function. Quadratic functions, characterized by their parabolic graphs, have either a maximum or minimum value, which directly influences their range. To determine the range of f(x) = -(x+5)(x+1), we need to identify whether the parabola opens upwards or downwards and find its vertex, which represents the maximum or minimum point.

Key Concepts to Remember:

  • Quadratic Function: A function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.
  • Parabola: The U-shaped curve that represents the graph of a quadratic function.
  • Vertex: The highest or lowest point on a parabola. It represents the maximum or minimum value of the function.
  • Range: The set of all possible output values (y-values) of a function.

Our mission is to find the range of the function f(x) = -(x+5)(x+1). To achieve this, we will follow a step-by-step approach:

  1. Expanding the Function: First, we expand the given function to express it in the standard quadratic form, which is f(x) = ax² + bx + c. This form will make it easier to identify the coefficients and determine the parabola's orientation.
  2. Determining the Parabola's Orientation: The coefficient of the x² term (a) dictates whether the parabola opens upwards (a > 0) or downwards (a < 0). This is critical because it tells us if the function has a minimum or maximum value.
  3. Finding the Vertex: The vertex is the turning point of the parabola and represents either the minimum or maximum value of the function. The x-coordinate of the vertex can be found using the formula x = -b / 2a. Once we have the x-coordinate, we can substitute it back into the function to find the y-coordinate, which represents the minimum or maximum value.
  4. Determining the Range: Based on the parabola's orientation and the vertex's y-coordinate, we can determine the range. If the parabola opens upwards, the range will be all real numbers greater than or equal to the y-coordinate of the vertex. If it opens downwards, the range will be all real numbers less than or equal to the y-coordinate of the vertex.

By systematically following these steps, we can accurately determine the range of the function f(x) = -(x+5)(x+1). This process not only helps us solve this specific problem but also provides a framework for analyzing the range of any quadratic function.

Step 1: Expanding the Function

To begin, let's expand the function f(x) = -(x+5)(x+1). This will transform it into the standard quadratic form, which is essential for further analysis. Expanding the expression, we get:

f(x) = -(x+5)(x+1) = -(x² + x + 5x + 5) = -(x² + 6x + 5) = -x² - 6x - 5

Now, the function is in the standard form f(x) = ax² + bx + c, where a = -1, b = -6, and c = -5. This form is crucial because the coefficients a, b, and c provide valuable information about the parabola's characteristics.

Step 2: Determining the Parabola's Orientation

The coefficient of the x² term, a, plays a pivotal role in determining the parabola's orientation. If a is positive (a > 0), the parabola opens upwards, indicating a minimum value. Conversely, if a is negative (a < 0), the parabola opens downwards, indicating a maximum value. In our case, a = -1, which is negative. Therefore, the parabola opens downwards. This tells us that the function f(x) = -x² - 6x - 5 has a maximum value, which will be the upper bound of its range.

Step 3: Finding the Vertex

The vertex is the turning point of the parabola and holds the key to determining the function's maximum or minimum value. The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

In our function, f(x) = -x² - 6x - 5, a = -1 and b = -6. Plugging these values into the formula, we get:

x = -(-6) / (2 * -1) = 6 / -2 = -3

So, the x-coordinate of the vertex is -3. Now, to find the y-coordinate, we substitute this value back into the function:

f(-3) = -(-3)² - 6(-3) - 5 = -9 + 18 - 5 = 4

Therefore, the vertex of the parabola is (-3, 4). Since the parabola opens downwards, the vertex represents the maximum point of the function, and the y-coordinate of the vertex, 4, is the maximum value of the function.

Step 4: Determining the Range

Now that we know the parabola opens downwards and the maximum value of the function is 4, we can determine the range. The range is the set of all possible output values (y-values) of the function. Since the parabola opens downwards, the function's values will be less than or equal to the maximum value. Therefore, the range of the function f(x) = -x² - 6x - 5 is:

All real numbers less than or equal to 4

This means that the function can produce any y-value that is 4 or smaller. There are no y-values greater than 4 in the function's output.

In this detailed exploration, we successfully determined the range of the quadratic function f(x) = -(x+5)(x+1). By expanding the function, identifying the parabola's orientation, finding the vertex, and understanding the relationship between these elements, we arrived at the conclusion that the range of the function is all real numbers less than or equal to 4. This comprehensive approach not only provides the answer to the specific question but also equips you with the knowledge and skills to analyze the range of any quadratic function. Understanding the range of a function is crucial in various mathematical and real-world applications, making this a valuable concept to master. The ability to determine the range of a quadratic function is a valuable skill in mathematics. By understanding the relationship between the quadratic equation, its parabolic graph, and the vertex, we can accurately determine the set of all possible output values. This knowledge is not only essential for solving mathematical problems but also for applying quadratic functions in real-world scenarios, such as optimization problems and modeling physical phenomena. Understanding these concepts solidifies a strong foundation in algebra and prepares you for more advanced mathematical topics. Remember, the key to mastering quadratic functions lies in understanding their properties and how they relate to their graphical representation. Keep practicing and exploring, and you'll be well on your way to mathematical proficiency!