Find The Interval Over Which The Graph Of The Function $f(x) = -(x+8)^2 - 1$ Is Decreasing.

Determining the intervals where a function is decreasing is a fundamental concept in calculus and precalculus mathematics. Understanding how a function's value changes as its input increases is crucial for analyzing its behavior and properties. In this comprehensive guide, we will delve into the function f(x) = -(x+8)^2 - 1, a quadratic function represented by a parabola, and meticulously explore the interval over which its graph is decreasing. We will begin by dissecting the key characteristics of parabolas, focusing on how the leading coefficient and vertex form influence their shape and direction. Then, we will apply these concepts to our specific function, identifying its vertex and axis of symmetry, which are crucial for determining the intervals of increase and decrease. By carefully analyzing the function's structure and utilizing visual aids, we will definitively pinpoint the interval where f(x) is decreasing, providing a step-by-step explanation that will solidify your understanding of this important mathematical concept.

Key Concepts: Parabolas and Decreasing Intervals

Before we dive into the specifics of f(x) = -(x+8)^2 - 1, it's essential to have a solid grasp of the underlying concepts. A parabola is the U-shaped curve that represents a quadratic function, which is a polynomial function of degree two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The coefficient a plays a pivotal role in determining the parabola's direction: if a is positive, the parabola opens upwards (a smile), and if a is negative, the parabola opens downwards (a frown). The vertex form of a quadratic function, f(x) = a(x-h)^2 + k, provides a more direct way to identify the vertex of the parabola, which is the point where the parabola changes direction. The vertex is represented by the coordinates (h, k). Understanding the vertex is crucial because it marks the boundary between the intervals where the function is increasing and decreasing.

A function is said to be decreasing over an interval if its values decrease as the input values increase. Visually, on a graph, this means the curve is sloping downwards as you move from left to right. For a parabola that opens downwards, like the one we'll be examining, the function increases up to the vertex and then decreases after the vertex. Therefore, to find the interval where f(x) is decreasing, we need to identify the vertex and focus on the x-values to the right of the vertex. This is because the function will be decreasing on the interval extending from the x-coordinate of the vertex to positive infinity. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = h, where h is the x-coordinate of the vertex. The axis of symmetry helps us visualize the symmetry of the parabola and further understand its increasing and decreasing behavior.

Now, let's apply these concepts to our specific function, f(x) = -(x+8)^2 - 1. This function is given in vertex form, which makes it easier to identify the key parameters. Comparing it to the general vertex form, f(x) = a(x-h)^2 + k, we can see that a = -1, h = -8, and k = -1. The negative value of a tells us that the parabola opens downwards, indicating that the function will have a maximum value at the vertex. The vertex coordinates are (h, k) = (-8, -1). This point represents the highest point on the graph of the function. The axis of symmetry is the vertical line x = -8, which divides the parabola into two symmetrical halves. To the left of this line, the function is increasing, and to the right, it is decreasing.

To further solidify our understanding, let's consider how the transformations affect the basic parabola y = x^2. The function f(x) = -(x+8)^2 - 1 can be seen as a series of transformations applied to the basic parabola. First, the term (x+8) represents a horizontal shift of 8 units to the left. Then, the negative sign in front of the parentheses reflects the parabola across the x-axis, causing it to open downwards. Finally, the term -1 represents a vertical shift of 1 unit downwards. These transformations combine to create the specific parabola we are analyzing. By understanding these transformations, we can better visualize the graph of the function and its key features.

Knowing that the parabola opens downwards and its vertex is at (-8, -1), we can now confidently determine the interval where the function is decreasing. Since the parabola opens downwards, it increases up to the vertex and then decreases after the vertex. Therefore, the function is decreasing for all x-values greater than -8. This interval can be represented in interval notation as (-8, ∞). It is important to note that the parenthesis indicates that -8 is not included in the interval, as the function is neither increasing nor decreasing at the vertex itself. The function transitions from increasing to decreasing at this point, making it a critical point of the function.

Based on our analysis, we've established that the function f(x) = -(x+8)^2 - 1 opens downwards and has a vertex at (-8, -1). This means the function increases until it reaches the vertex and then begins to decrease. To find the interval where the function is decreasing, we need to consider the x-values to the right of the vertex. The x-coordinate of the vertex is -8, so the function will be decreasing for all x-values greater than -8. In interval notation, this is represented as (-8, ∞). This interval represents all real numbers greater than -8, extending infinitely to the right on the number line.

To visualize this, imagine tracing the graph of the parabola from left to right. As you move along the curve, the function's values increase until you reach the vertex at (-8, -1). After this point, the function's values start to decrease as you continue moving to the right. The graph slopes downwards, indicating a decreasing function. This decreasing behavior continues indefinitely as x increases, which is why the interval extends to positive infinity. It is essential to remember that the function is not decreasing at the vertex itself. The vertex is a point of transition, where the function changes from increasing to decreasing. Therefore, -8 is not included in the interval of decrease, and we use a parenthesis to denote this exclusion.

In summary, the interval over which the graph of f(x) = -(x+8)^2 - 1 is decreasing is (-8, ∞). This means that for any x-value greater than -8, the function's value will be less than the function's value at any smaller x-value within this interval. Understanding the relationship between the vertex, the direction of the parabola, and the concept of decreasing intervals is crucial for analyzing quadratic functions and their graphs. This knowledge provides a foundation for solving more complex problems in calculus and related fields.

A visual representation of the function f(x) = -(x+8)^2 - 1 can significantly enhance our understanding of the decreasing interval. By sketching the graph of the parabola, we can see the relationship between the function's behavior and its corresponding x-values. The parabola opens downwards, confirming our earlier analysis based on the negative leading coefficient. The vertex, located at (-8, -1), is the highest point on the graph. To the left of the vertex, the parabola slopes upwards, indicating that the function is increasing. To the right of the vertex, the parabola slopes downwards, illustrating the decreasing behavior of the function.

By focusing on the portion of the graph to the right of the vertex, we can clearly see the decreasing interval. As we move along the x-axis from -8 towards positive infinity, the y-values of the function continuously decrease. This downward slope visually confirms that the function is decreasing over the interval (-8, ∞). The graph provides a concrete representation of the abstract concept of decreasing intervals, making it easier to grasp the relationship between the function's values and its input values. Furthermore, the graph highlights the symmetry of the parabola around the axis of symmetry, which passes through the vertex. This symmetry reinforces the understanding that the function's behavior on one side of the vertex is a mirror image of its behavior on the other side. However, it is crucial to note that the decreasing interval only considers the x-values to the right of the vertex, as this is the region where the function is sloping downwards.

In addition to the basic sketch, we can also use graphing software or online tools to create a more precise representation of the function. These tools allow us to zoom in on specific regions of the graph and observe the function's behavior in greater detail. By plotting additional points on the graph, we can further confirm the decreasing trend over the interval (-8, ∞). Visualizing the function through a graph provides a powerful tool for understanding and analyzing its properties, including its intervals of increase and decrease. It bridges the gap between the algebraic representation of the function and its geometric interpretation, fostering a deeper understanding of the mathematical concepts involved.

In conclusion, after a thorough analysis of the function f(x) = -(x+8)^2 - 1, we have definitively determined that the graph of the function is decreasing over the interval (-8, ∞). This conclusion is based on our understanding of parabolas, their vertex form, and the concept of decreasing intervals. We identified the key parameters of the function, including the vertex at (-8, -1) and the fact that the parabola opens downwards due to the negative leading coefficient. These factors allowed us to deduce that the function increases up to the vertex and then decreases for all x-values greater than -8.

We further reinforced our understanding by visualizing the function's graph, which clearly illustrates the decreasing behavior to the right of the vertex. The graph provides a tangible representation of the abstract concept of decreasing intervals, making it easier to grasp the relationship between the function's values and its input values. By connecting the algebraic analysis with the visual representation, we have gained a comprehensive understanding of the function's behavior. This understanding is crucial for solving a wide range of mathematical problems, including optimization problems, curve sketching, and calculus applications. The ability to identify intervals of increase and decrease is a fundamental skill in mathematics, and this comprehensive guide has provided a step-by-step approach to mastering this skill.

The interval (-8, ∞) represents all real numbers greater than -8, excluding -8 itself. This means that for any x-value within this interval, the function's value will be less than its value at any smaller x-value within the same interval. This decreasing behavior is a direct consequence of the parabola opening downwards and the vertex marking the transition point from increasing to decreasing. By understanding these principles, we can confidently analyze other quadratic functions and determine their intervals of increase and decrease. This knowledge empowers us to make predictions about the function's behavior and solve a variety of mathematical problems efficiently and accurately.