Decreasing Interval Of Transformed Absolute Value Function G(x) = |x+1|-7

In this article, we will delve into the transformation of the absolute value function and identify the interval where the transformed function is decreasing. The original function is f(x) = |x|, and it undergoes transformations to become g(x) = |x + 1| - 7. Our primary goal is to determine the interval on which the function g(x) is decreasing. To achieve this, we will first discuss the properties of the absolute value function, then analyze the transformations applied, and finally, pinpoint the interval of decrease.

Understanding the Absolute Value Function

The absolute value function, denoted as f(x) = |x|, is a fundamental concept in mathematics. Its basic form represents the distance of a number x from zero on the number line. The graph of f(x) = |x| is a V-shaped graph with its vertex at the origin (0, 0). The function is defined as:

• f(x) = x when x ≥ 0
• f(x) = -x when x < 0

This piecewise definition indicates that for any non-negative value of x, the function returns the value itself. For any negative value of x, the function returns the opposite (positive) value. Key features of the absolute value function include:

• Symmetry: The graph is symmetric about the y-axis, meaning that the function is an even function (f(x) = f(-x)).
• Vertex: The vertex, or turning point, of the graph is at the origin (0, 0).
• Decreasing Interval: The function is decreasing on the interval (-∞, 0]. As x approaches zero from the left, f(x) decreases.
• Increasing Interval: The function is increasing on the interval [0, ∞). As x moves away from zero towards positive infinity, f(x) increases.

Understanding these basic characteristics is crucial for analyzing how transformations affect the behavior of the absolute value function. By knowing the original shape and behavior, we can more easily predict how the transformed function will behave. The decreasing interval of the parent function, f(x) = |x|, is (-∞, 0], which serves as a baseline for our analysis of the transformed function, g(x) = |x + 1| - 7. This foundation helps us to systematically approach the problem by considering how horizontal and vertical shifts impact the decreasing interval. The symmetry of the absolute value function further simplifies our analysis, as we can focus on how the vertex is shifted, and the rest of the graph will follow symmetrically. The understanding of intervals where the function increases or decreases is vital in calculus and other advanced mathematical concepts. Therefore, a solid grasp of these properties is essential for further studies in mathematics. The next step in our analysis is to examine the transformations applied to the original function to obtain g(x), which will allow us to determine the new interval of decrease.

Analyzing the Transformations

The given transformation takes the original function f(x) = |x| and transforms it into g(x) = |x + 1| - 7. This transformation involves two key steps: a horizontal shift and a vertical shift. Understanding these shifts is essential for determining how the decreasing interval changes.

1. Horizontal Shift: The term (x + 1) inside the absolute value function indicates a horizontal shift. Specifically, replacing x with (x + 1) shifts the graph 1 unit to the left. This is because the function now behaves as if the y-axis has been moved 1 unit to the right. Consequently, the vertex of the graph, which was originally at (0, 0), is shifted to (-1, 0).

   • The transformation x → (x + 1) represents a shift to the left along the x-axis.
   • The new vertex after this shift is at x = -1.

   This shift affects the interval where the function decreases. Since the entire graph moves 1 unit to the left, the interval of decrease also shifts 1 unit to the left. The original decreasing interval (-∞, 0] now becomes (-∞, -1]. This horizontal shift is a crucial element in understanding the overall transformation, as it directly impacts the position of the vertex and, consequently, the decreasing and increasing intervals of the function. This concept of horizontal transformations is fundamental in function analysis and is applicable across various types of functions, not just absolute value functions. The key idea is to recognize that adding a constant inside the function's argument (in this case, x) will result in a horizontal shift, with the direction of the shift being opposite to the sign of the constant. Thus, (x + 1) shifts the graph to the left, while (x - 1) would shift it to the right.

2. Vertical Shift: The term -7 outside the absolute value function indicates a vertical shift. Subtracting 7 from the function shifts the entire graph 7 units downward. This means the vertex, which was at (-1, 0) after the horizontal shift, is now at (-1, -7).

   • The transformation f(x) → f(x) - 7 represents a shift downward along the y-axis.
   • The new vertex after this shift is at y = -7.

   The vertical shift does not affect the intervals of increase or decrease. It only changes the vertical position of the graph. Therefore, the decreasing interval remains the same after this shift. However, the vertical shift is important in determining the overall shape and position of the graph, which can be useful for other analyses, such as finding the range of the function or identifying key points. The vertical shift is a straightforward transformation; adding or subtracting a constant outside the function directly moves the graph up or down, respectively. Understanding vertical shifts, along with horizontal shifts, provides a complete picture of how transformations can alter the position and shape of a graph. In the context of g(x) = |x + 1| - 7, the vertical shift complements the horizontal shift, resulting in a final vertex position of (-1, -7). This new vertex is the key to determining the interval where the transformed function is decreasing.

By understanding these transformations, we can accurately determine the interval where the function g(x) = |x + 1| - 7 is decreasing. The horizontal shift moves the interval, while the vertical shift simply repositions the graph vertically without affecting the intervals of increase or decrease. Therefore, the combined effect of these transformations allows us to pinpoint the specific interval where g(x) exhibits decreasing behavior.

Determining the Decreasing Interval

Based on the transformation analysis, we know that the function g(x) = |x + 1| - 7 is the result of shifting the original absolute value function f(x) = |x| one unit to the left and seven units down. The horizontal shift is the critical factor in determining the new interval of decrease.

The original function f(x) = |x| decreases on the interval (-∞, 0]. The horizontal shift of 1 unit to the left transforms this interval. To find the new interval, we consider how the x-coordinate of the vertex changes. The original vertex at (0, 0) is shifted to (-1, -7). Therefore, the new decreasing interval starts from negative infinity and ends at the x-coordinate of the new vertex.

The horizontal shift (x + 1) moves the entire graph one unit to the left. Consequently, the interval where the function decreases also shifts one unit to the left. Thus, the decreasing interval for g(x) becomes (-∞, -1]. This means that as x approaches -1 from the left, the function g(x) is decreasing. Beyond x = -1, the function starts to increase due to the nature of the absolute value function.

The vertical shift of 7 units down does not affect the decreasing interval. Vertical shifts only move the graph up or down, changing the y-coordinates but not the x-coordinates. Therefore, the decreasing interval remains unchanged by the vertical shift. The focus remains solely on the horizontal shift to accurately determine the interval where the function decreases.

Therefore, the function g(x) = |x + 1| - 7 is decreasing on the interval (-∞, -1]. This is because the graph of the absolute value function is V-shaped, and after shifting the vertex to (-1, -7), the left side of the V (i.e., x < -1) represents the decreasing portion of the function. The right side of the V (x > -1) represents the increasing portion. The ability to identify such intervals is essential in various mathematical contexts, including optimization problems, where understanding function behavior is critical for finding maximum and minimum values. The analysis here underscores the importance of understanding how transformations affect the key characteristics of functions, such as intervals of increase and decrease. By systematically breaking down the transformations, we can accurately determine the new intervals for the transformed function.

Conclusion

In conclusion, the function g(x) = |x + 1| - 7 is decreasing on the interval (-∞, -1]. This was determined by analyzing the transformations applied to the original absolute value function f(x) = |x|. The horizontal shift of 1 unit to the left is the critical factor in changing the decreasing interval, while the vertical shift of 7 units down does not affect the interval of decrease. Understanding the properties of absolute value functions and the effects of transformations is essential for accurately determining the intervals where the function is decreasing or increasing. This analysis provides a comprehensive approach to solving such problems and highlights the importance of considering each transformation step by step. The key takeaway is that horizontal shifts directly impact the intervals of increase and decrease, while vertical shifts only affect the vertical position of the graph. By applying these principles, we can confidently analyze and understand the behavior of transformed functions.