Shifting The Graph Of H(x) = -2/3x + 5 Down 3 Units A Comprehensive Guide

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#h1 Consider the Graph of the Linear Function h(x) = -2/3x + 5 A Comprehensive Guide

This article delves into the intricacies of the linear function h(x) = -2/3x + 5, exploring its graphical representation and the transformations that can be applied to it. Specifically, we will address the question of how to shift the graph of this function downwards by 3 units. To fully understand this, we will first break down the components of a linear function and then discuss the effect of changing these components on the graph. This analysis will not only help answer the given question but also provide a solid foundation for understanding linear functions and their transformations in general. Our goal is to provide a clear and concise explanation, making it accessible to anyone interested in learning more about linear functions, whether you are a student, an educator, or simply someone with a curiosity for mathematics. By the end of this article, you will have a comprehensive understanding of how to manipulate the graph of a linear function, specifically how to shift it vertically, and you'll be equipped with the knowledge to tackle similar problems with confidence. The concepts discussed here are fundamental in algebra and calculus, making this a valuable read for anyone looking to strengthen their mathematical skills.

Understanding Linear Functions

Linear functions are a cornerstone of algebra, and understanding their properties is crucial for success in higher-level mathematics. A linear function can be generally represented in the slope-intercept form as y = mx + b, where m represents the slope and b represents the y-intercept. In the given function, h(x) = -2/3x + 5, the slope m is -2/3, and the y-intercept b is 5. The slope indicates the steepness and direction of the line. A negative slope, like -2/3, means the line slopes downwards from left to right. The y-intercept is the point where the line crosses the y-axis, and in this case, it's at the point (0, 5). Visualizing a linear function on a graph is essential. The graph is a straight line that extends infinitely in both directions. Each point on the line represents a solution to the equation h(x) = -2/3x + 5. To plot the graph, you can start by plotting the y-intercept (0, 5) and then use the slope to find another point. Since the slope is -2/3, for every 3 units you move to the right on the x-axis, you move 2 units down on the y-axis. This allows you to plot multiple points and draw a straight line through them. Understanding the relationship between the equation and the graph is key to manipulating linear functions. Changes to the slope or y-intercept will result in changes to the graph, and vice versa. By grasping these fundamental concepts, you can predict how the graph will change based on alterations to the equation. This knowledge is not only useful for solving problems but also for understanding real-world applications of linear functions, such as modeling relationships between variables in economics, physics, and other fields.

The Role of the Y-Intercept

The y-intercept, denoted by b in the slope-intercept form y = mx + b, plays a pivotal role in determining the vertical position of a linear function's graph. It is the point where the line intersects the y-axis, which occurs when x = 0. In the given function, h(x) = -2/3x + 5, the y-intercept is 5, indicating that the line crosses the y-axis at the point (0, 5). Changing the y-intercept directly affects the vertical shift of the graph. Increasing the y-intercept shifts the graph upwards, while decreasing it shifts the graph downwards. This is because the y-intercept essentially sets the baseline vertical position of the line. For instance, if we were to change the y-intercept from 5 to 8, the entire line would shift 3 units upwards. Conversely, if we changed it to 2, the line would shift 3 units downwards. This vertical shift is uniform across the entire line, meaning that every point on the line moves the same distance vertically. Understanding this direct relationship between the y-intercept and the vertical position of the graph is crucial for solving problems involving vertical translations of linear functions. In the context of the question, which asks how to move the graph down 3 units, the key is to adjust the y-intercept. By subtracting 3 from the original y-intercept of 5, we can achieve the desired downward shift. This concept is not only applicable to linear functions but also extends to other types of functions, making it a fundamental principle in understanding function transformations. By mastering the role of the y-intercept, you gain a powerful tool for manipulating and interpreting linear functions and their graphs.

Shifting the Graph Downward

To shift the graph of the linear function h(x) = -2/3x + 5 down by 3 units, we need to adjust the y-intercept. As discussed earlier, the y-intercept determines the vertical position of the graph. To move the graph down, we need to decrease the y-intercept. The original y-intercept in the function is 5. To shift the graph down 3 units, we subtract 3 from the original y-intercept: 5 - 3 = 2. Therefore, the new y-intercept should be 2. This means the new function, which represents the shifted graph, will have the same slope (-2/3) but a y-intercept of 2. The equation of the new function would be h'(x) = -2/3x + 2, where h'(x) represents the transformed function. It's important to note that changing the slope would alter the steepness and direction of the line, but it wouldn't simply shift the graph vertically. Only changes to the y-intercept result in a vertical translation. Visualizing this transformation on a graph can be helpful. Imagine the original line and then visualize sliding it down 3 units along the y-axis. The new line will be parallel to the original line but positioned lower on the graph. Each point on the original line will have moved 3 units down to its corresponding point on the new line. This concept of vertical translation by adjusting the y-intercept is a fundamental principle in function transformations and is widely applicable in various mathematical contexts. By understanding this principle, you can easily manipulate the graphs of linear functions and other types of functions to achieve desired transformations.

Analyzing the Options

Now, let's analyze the given options in the context of shifting the graph of h(x) = -2/3x + 5 down 3 units:

  • A. the value of b to -3: This option suggests changing the y-intercept (b) to -3. If we substitute -3 for b in the equation, we get h(x) = -2/3x - 3. This would shift the graph down significantly more than 3 units (specifically, 8 units down, since 5 - (-3) = 8). Therefore, this option is incorrect.
  • B. the value of m to -3: This option suggests changing the slope (m) to -3. Changing the slope would alter the steepness and direction of the line, but it wouldn't simply shift the graph vertically. It would rotate the line, changing its angle with the x-axis. Therefore, this option is incorrect.
  • C. the value of b to 2: This option suggests changing the y-intercept (b) to 2. As we discussed earlier, to shift the graph down 3 units, we need to subtract 3 from the original y-intercept: 5 - 3 = 2. This option correctly identifies the new y-intercept needed for the desired transformation. Therefore, this option is correct.
  • D. the value of m to the Discussion category: This option seems incomplete and doesn't provide a clear mathematical operation. It mentions the discussion category, which is not relevant to changing the graph of a linear function. Therefore, this option is incorrect.

By carefully analyzing each option and applying our understanding of the role of the y-intercept in vertical translations, we can confidently conclude that the correct option is C. Changing the value of b to 2 will shift the graph of h(x) = -2/3x + 5 down 3 units.

Conclusion

In conclusion, to shift the graph of the linear function h(x) = -2/3x + 5 down by 3 units, the correct approach is to change the value of the y-intercept (b) to 2. This is because the y-intercept directly controls the vertical position of the graph, and decreasing it by 3 will result in the desired downward shift. We arrived at this conclusion by first understanding the fundamental properties of linear functions, particularly the slope-intercept form (y = mx + b) and the roles of the slope (m) and y-intercept (b). We then focused on the specific effect of changing the y-intercept, which causes a vertical translation of the graph. By subtracting 3 from the original y-intercept of 5, we determined the new y-intercept should be 2. We also analyzed the given options, eliminating those that involved changing the slope or suggested an incorrect value for the y-intercept. This process highlights the importance of understanding the relationship between the equation of a linear function and its graphical representation. By mastering these concepts, you can confidently manipulate linear functions and their graphs to achieve desired transformations. This knowledge is not only valuable for solving mathematical problems but also for understanding real-world applications of linear functions in various fields. The ability to visualize and manipulate graphs is a fundamental skill in mathematics, and this article has provided a comprehensive guide to achieving this for linear functions. We hope this detailed explanation has clarified the concepts and equipped you with the tools to tackle similar problems with ease.