Transforming Absolute Value Functions: A Visual Guide

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Transforming Absolute Value Functions: A Visual Guide

Understanding transformations of functions is a fundamental concept in mathematics. In this guide, we'll explore how to transform the graph of one absolute value function into another. Specifically, we'll investigate the transformation that converts the graph of f(x) = 10|x| - 8 into the graph of g(x) = 5|x| - 8. Let's dive in and unravel the mystery behind this transformation!

Understanding the Functions

Before we jump into the transformation, let's take a closer look at the two functions involved:

  • f(x) = 10|x| - 8: This is an absolute value function where the absolute value of x is multiplied by 10, and then 8 is subtracted. The "10" affects the steepness of the V-shape, and the "-8" shifts the entire graph downward.
  • g(x) = 5|x| - 8: This is also an absolute value function, but here, the absolute value of x is multiplied by 5, and then 8 is subtracted. Notice that the "-8" is the same as in f(x), but the multiplier of the absolute value is different. This difference in the multiplier is key to understanding the transformation.

Understanding these components helps us visualize how one graph morphs into the other. The constant term (-8) determines the vertical position of the vertex of the absolute value function, while the coefficient of the absolute value term dictates the rate at which the function increases or decreases as we move away from the vertex along the x-axis. A larger coefficient results in a steeper slope and a narrower V-shaped graph, while a smaller coefficient leads to a gentler slope and a wider V-shaped graph. This distinction is crucial in identifying whether the transformation involves stretching or shrinking the graph, either horizontally or vertically. Analyzing these aspects allows us to predict and interpret the changes in the shape and position of the graph as we transition from f(x) to g(x), providing valuable insights into the nature of the transformation.

Identifying the Transformation

Now, let's figure out what kind of transformation takes f(x) to g(x). We're essentially going from 10|x| - 8 to 5|x| - 8. The "-8" remains constant, meaning there's no vertical shift difference between the two after accounting for the initial vertical shift. However, the coefficient of the absolute value changes from 10 to 5. This change affects the vertical scaling of the function. Since the coefficient is decreasing, the graph is being compressed or shrunk in the vertical direction.

To further clarify, let's consider what happens when we apply different types of transformations. A horizontal shrink would compress the graph towards the y-axis, making it narrower. A horizontal stretch would pull the graph away from the y-axis, making it wider. A vertical stretch would elongate the graph vertically, making it taller, while a vertical shrink would compress the graph vertically, making it shorter. In our case, since the coefficient of the absolute value term is decreasing, the graph becomes less steep, indicating a vertical compression. This means that for any given x-value, the corresponding y-value in g(x) is closer to the x-axis than the y-value in f(x). Therefore, the transformation that converts the graph of f(x) = 10|x| - 8 into the graph of g(x) = 5|x| - 8 is a vertical shrink, making the graph wider and less steep. This conclusion aligns with our understanding of how vertical transformations affect the shape of absolute value functions, providing a clear and logical explanation for the observed change.

Vertical Shrink Explained

A vertical shrink (also known as a vertical compression) makes the graph shorter. In our case, to go from f(x) = 10|x| - 8 to g(x) = 5|x| - 8, we're essentially halving the coefficient of the absolute value term (from 10 to 5). This means that for any given x value, the y value of g(x) (after accounting for the vertical shift of -8) is half of what it was in f(x) (after accounting for the vertical shift of -8). Therefore, the graph is compressed vertically toward the x-axis, making it wider. The vertex remains at the same x-coordinate because the horizontal position isn't altered; however, the slopes of the lines forming the absolute value shape are less steep.

Understanding the mechanism of a vertical shrink involves grasping how it alters the vertical distances on the graph. When we apply a vertical shrink by a factor of k, where 0 < k < 1, each y-value on the original graph is multiplied by k. This multiplication compresses the graph towards the x-axis, effectively reducing the vertical height of the graph. In the context of absolute value functions, a vertical shrink affects the slopes of the linear segments that form the V-shape. Specifically, if we have a function f(x) = a|x|, applying a vertical shrink by a factor of k results in a new function g(x) = ka*|x|*. The slopes of the linear segments in g(x) are k times the slopes in f(x), making the graph wider and less steep. This change in slope is a direct consequence of the vertical compression, illustrating how the transformation alters the rate of change of the function with respect to x. Thus, a vertical shrink not only reduces the vertical dimensions of the graph but also modifies its overall shape by adjusting the steepness of its components, leading to a visually distinct transformation.

Why Not the Other Options?

Let's quickly eliminate the other options to reinforce our understanding:

  • Horizontal Shrink: This would compress the graph horizontally, making it narrower. This isn't happening here.
  • Vertical Stretch: This would stretch the graph vertically, making it taller. The opposite is occurring.
  • Horizontal Stretch: This would stretch the graph horizontally, making it wider. While the graph does become wider, it's a result of the vertical shrink, not a horizontal stretch directly.

To elaborate further on why these options are incorrect, let's consider how each transformation would affect the original function f(x) = 10|x| - 8. A horizontal shrink would involve replacing x with cx, where c > 1, resulting in f(cx) = 10|cx| - 8. This would compress the graph horizontally, making it narrower, which is not what we observe in the transformation from f(x) to g(x). A vertical stretch would involve multiplying the entire function by a factor k, where k > 1, resulting in kf(x) = k(10|x| - 8) = 10k|x| - 8k. This would stretch the graph vertically, making it taller, which is also contrary to the observed transformation. A horizontal stretch would involve replacing x with x/c, where c > 1, resulting in f(x/c) = 10|x/c| - 8. This would stretch the graph horizontally, making it wider. However, the widening of the graph from f(x) to g(x) is a consequence of the vertical compression, not a direct horizontal stretch. The key difference lies in the fact that the coefficient of the absolute value term is changing, indicating a vertical transformation rather than a horizontal one. Therefore, understanding the specific effects of each type of transformation allows us to confidently rule out these options and focus on the vertical shrink as the correct answer.

Conclusion

The correct answer is D. vertical shrink. The graph of f(x) = 10|x| - 8 is transformed into the graph of g(x) = 5|x| - 8 by a vertical shrink, compressing the graph vertically and making it wider.

Understanding function transformations is crucial for mastering many areas of mathematics. By recognizing how changes to the equation affect the graph, you can quickly visualize and analyze different functions. So, keep practicing and exploring different transformations to build your mathematical intuition!

In summary, mastering function transformations involves understanding how changes to the equation affect the graph's shape and position. By recognizing the specific effects of vertical and horizontal stretches, shrinks, and shifts, you can quickly visualize and analyze different functions. Remember that vertical transformations affect the y-values of the function, while horizontal transformations affect the x-values. A vertical shrink compresses the graph towards the x-axis, making it wider and less steep, while a vertical stretch elongates the graph vertically, making it taller and steeper. Horizontal stretches and shrinks, on the other hand, affect the width of the graph by compressing or expanding it along the x-axis. By practicing and exploring various transformations, you can develop a strong intuition for how functions behave and build a solid foundation for more advanced mathematical concepts. So, keep exploring and experimenting with different transformations to deepen your understanding and enhance your problem-solving skills!