Determining Decreasing Interval Of F(x) = -(x+8)^2 - 1

Introduction

In the realm of mathematics, understanding the behavior of functions is crucial. Specifically, identifying intervals where a function is increasing or decreasing provides valuable insights into its overall characteristics. This article will delve into the process of determining the decreasing interval of a quadratic function, using the example of f(x) = -(x+8)^2 - 1. We will explore the properties of quadratic functions, their graphical representation as parabolas, and how the vertex and leading coefficient influence the function's increasing and decreasing behavior. By the end of this discussion, you will be equipped with the knowledge to confidently identify decreasing intervals for various quadratic functions.

Understanding Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. These functions are characterized by their parabolic graphs, which are U-shaped curves. The parabola opens upwards if the leading coefficient (a) is positive and downwards if a is negative. The vertex of the parabola is the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

The given function, f(x) = -(x+8)^2 - 1, is a quadratic function in vertex form. The vertex form of a quadratic function is f(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex. In our case, a = -1, h = -8, and k = -1. This tells us that the parabola opens downwards (since a is negative) and the vertex is located at the point (-8, -1). The vertex plays a crucial role in determining the intervals where the function is increasing or decreasing. To fully grasp this, let's dissect the function's structure. The term (x+8)^2 will always be non-negative since it's a square. The negative sign in front of it, - (x+8)^2, makes this term non-positive. Subtracting 1, - (x+8)^2 - 1, further shifts the function downwards. Therefore, the maximum value of this function occurs at the vertex (-8, -1), and the function decreases as we move away from this vertex in either direction along the x-axis. Understanding the vertex form is crucial for visualizing how transformations like shifts and reflections impact the basic parabola y = x^2. By recognizing these transformations, we can quickly deduce the behavior of the function without needing to perform extensive calculations or plotting numerous points.

Determining the Decreasing Interval

To determine where a function is decreasing, we need to identify the interval(s) on the x-axis where the function's values are getting smaller as x increases. For a parabola that opens downwards, the function is increasing to the left of the vertex and decreasing to the right of the vertex. In our example, f(x) = -(x+8)^2 - 1, the vertex is at (-8, -1) and the parabola opens downwards. This means that the function is increasing on the interval (-∞, -8) and decreasing on the interval (-8, ∞). Visually, imagine walking along the parabola from left to right. As you approach the vertex from the left, you are climbing uphill, indicating an increasing function. Once you pass the vertex, you start walking downhill, signifying a decreasing function. The x-coordinate of the vertex, in this case -8, is the critical point that separates the increasing and decreasing intervals. It's the point where the function transitions from going up to going down. Therefore, the decreasing interval is the set of all x-values greater than -8. This can be written in interval notation as (-8, ∞). It's important to note that we use an open interval here because the function is neither increasing nor decreasing at the vertex itself; it's momentarily stationary at that point. The concept of increasing and decreasing intervals is fundamental in calculus, where we use derivatives to formally define and calculate these intervals. However, for quadratic functions, we can often determine these intervals simply by understanding the shape of the parabola and the location of its vertex.

Analyzing the Graph

The graph of f(x) = -(x+8)^2 - 1 is a parabola that opens downwards with its vertex at (-8, -1). As we move from left to right along the x-axis, the function's values increase until we reach x = -8. At this point, the function attains its maximum value of -1. As we continue moving to the right (i.e., for x > -8), the function's values decrease. This decreasing behavior is evident in the downward slope of the parabola to the right of the vertex. To visualize this, you can imagine drawing a tangent line to the parabola at various points. To the left of the vertex, these tangent lines will have positive slopes, indicating an increasing function. To the right of the vertex, the tangent lines will have negative slopes, indicating a decreasing function. At the vertex itself, the tangent line will be horizontal, with a slope of zero. This graphical representation provides a clear and intuitive way to understand the concept of increasing and decreasing intervals. By analyzing the graph, we can confirm that the function f(x) = -(x+8)^2 - 1 is decreasing on the interval (-8, ∞). This visual approach is particularly helpful for students who are still developing their understanding of functions and their properties. The graph serves as a concrete representation of the abstract concept of increasing and decreasing behavior.

Conclusion

In conclusion, the graph of f(x) = -(x+8)^2 - 1 is decreasing on the interval (-8, ∞). This was determined by analyzing the function's vertex form, understanding that the parabola opens downwards, and recognizing that the decreasing interval lies to the right of the vertex. The vertex, being the maximum point of the parabola, acts as a turning point, transitioning the function from increasing to decreasing behavior. Understanding the interplay between the vertex, the leading coefficient, and the overall shape of the parabola is key to identifying these intervals. This knowledge is fundamental in mathematics, particularly in calculus, where the concepts of increasing and decreasing functions are crucial for optimization problems and curve sketching. By mastering these concepts, you can gain a deeper understanding of the behavior of functions and their applications in various fields. Moreover, the ability to analyze functions and determine their increasing and decreasing intervals is a valuable skill that extends beyond the classroom and into real-world problem-solving scenarios.

Therefore, the correct answer is C. (-8, ∞].