Identifying Decreasing Intervals Of Transformed Absolute Value Functions
In the realm of mathematics, grasping the behavior of functions is paramount. This article delves into the transformation of the absolute value function, specifically f(x) = |x|, and its subsequent shift to g(x) = |x + 1| – 7. Our primary focus will be on identifying the interval over which this transformed function, g(x), is decreasing. To achieve this, we'll embark on a comprehensive exploration of absolute value functions, their transformations, and the concept of decreasing intervals. Understanding these concepts is crucial for students, educators, and anyone with a passion for mathematical analysis. This article aims to provide a clear and detailed explanation, making the topic accessible to a wide audience. By the end of this discussion, you will have a solid understanding of how transformations affect the behavior of absolute value functions and how to determine their intervals of decrease.
At the heart of our discussion lies the absolute value function, denoted as f(x) = |x|. This function, while seemingly simple, exhibits a unique characteristic: it returns the non-negative magnitude of any real number input. In simpler terms, the absolute value of a number is its distance from zero on the number line, irrespective of direction. Mathematically, this is defined piecewise:
- f(x) = x, if x ≥ 0
- f(x) = -x, if x < 0
The graph of f(x) = |x| is a distinctive V-shape, with the vertex (the point where the two lines meet) situated at the origin (0, 0). The right arm of the V corresponds to the portion where x is non-negative, and the left arm corresponds to the portion where x is negative. This V-shape is a direct consequence of the function's definition: for positive x, the function simply returns x, resulting in a line with a slope of 1; for negative x, the function returns the negation of x, resulting in a line with a slope of -1. The symmetry of the graph about the y-axis reflects the fact that the absolute value of a number and its negation are the same. This fundamental understanding of the absolute value function's behavior and graphical representation is crucial for analyzing its transformations.
Transformations of Functions: A General Overview
Before diving into the specific transformations applied to our absolute value function, it's beneficial to establish a general understanding of function transformations. Transformations are operations that alter the graph of a function, changing its position, shape, or size. These transformations can be broadly categorized into:
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Vertical and Horizontal Shifts (Translations): These transformations involve moving the entire graph up, down, left, or right without changing its shape. A vertical shift is achieved by adding or subtracting a constant from the function's output, while a horizontal shift is achieved by adding or subtracting a constant from the function's input.
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Vertical and Horizontal Stretches and Compressions (Scales): These transformations alter the graph's size either vertically or horizontally. A vertical stretch or compression is achieved by multiplying the function's output by a constant, while a horizontal stretch or compression is achieved by multiplying the function's input by a constant.
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Reflections: These transformations flip the graph across an axis. Reflection across the x-axis is achieved by negating the function's output, while reflection across the y-axis is achieved by negating the function's input.
Understanding how these transformations affect the graph of a function is essential for analyzing and manipulating functions effectively. In our case, we'll focus on horizontal and vertical shifts as they are the transformations applied to f(x) = |x| to obtain g(x) = |x + 1| – 7.
Transforming f(x) = |x| to g(x) = |x + 1| – 7
Now, let's dissect the transformation of f(x) = |x| into g(x) = |x + 1| – 7. This transformation involves two distinct operations:
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Horizontal Shift: The term |x + 1| indicates a horizontal shift. Specifically, adding 1 to the input x results in a shift of the graph 1 unit to the left. This might seem counterintuitive, but it's crucial to remember that the transformation affects the input value. To understand why it shifts left, consider what value of x makes the expression inside the absolute value zero. For |x|, it's x = 0. For |x + 1|, it's x = -1. Thus, the vertex of the transformed function is shifted to x = -1.
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Vertical Shift: The term – 7 outside the absolute value represents a vertical shift. Subtracting 7 from the output of the absolute value function shifts the entire graph 7 units downward. This is a more straightforward transformation, as it directly affects the output value of the function.
Combining these two transformations, we can visualize the graph of g(x) = |x + 1| – 7 as the graph of f(x) = |x| shifted 1 unit to the left and 7 units downward. The vertex of the V-shape, which was initially at (0, 0) for f(x), is now located at (-1, -7) for g(x). This shift in the vertex is critical for determining the intervals where the function is increasing or decreasing.
Determining Intervals of Increase and Decrease
To determine the intervals where a function is increasing or decreasing, we need to analyze its behavior as x increases. A function is said to be increasing on an interval if its output values increase as its input values increase. Conversely, a function is said to be decreasing on an interval if its output values decrease as its input values increase. Graphically, this corresponds to the function's graph sloping upwards (increasing) or downwards (decreasing) as we move from left to right.
For the absolute value function g(x) = |x + 1| – 7, the key to determining the intervals of increase and decrease lies in the location of the vertex. As we've established, the vertex of g(x) is at (-1, -7). To the left of the vertex, the graph slopes downwards, indicating that the function is decreasing. To the right of the vertex, the graph slopes upwards, indicating that the function is increasing. Therefore, the function g(x) = |x + 1| – 7 is decreasing on the interval (-∞, -1) and increasing on the interval (-1, ∞). The value x = -1 is the point where the function transitions from decreasing to increasing, and it corresponds to the x-coordinate of the vertex.
The Interval Where g(x) = |x + 1| – 7 is Decreasing
Based on our analysis, we can definitively state that the function g(x) = |x + 1| – 7 is decreasing on the interval (-∞, -1). This interval encompasses all real numbers less than -1. As we move along the x-axis from negative infinity towards -1, the corresponding y-values of the function decrease. This is because, to the left of the vertex at x = -1, the graph of the absolute value function has a negative slope. The further we move to the left from -1, the lower the function's value becomes.
Conclusion
In this article, we've explored the transformation of the absolute value function f(x) = |x| into g(x) = |x + 1| – 7. We've identified the horizontal and vertical shifts that constitute this transformation and understood how they affect the graph of the function. Furthermore, we've determined the interval where the transformed function is decreasing. Our analysis has revealed that g(x) = |x + 1| – 7 is decreasing on the interval (-∞, -1). This conclusion is drawn from understanding the shape of the absolute value function, the impact of transformations on its vertex, and the relationship between the graph's slope and the function's increasing or decreasing behavior. This comprehensive understanding of function transformations and intervals of increase and decrease is invaluable in various mathematical contexts and applications.