Transforming Graphs Finding The Rule For Y = 2x + 7 To Y = 3x + 2
Introduction
In the fascinating world of mathematics, transformations play a pivotal role in altering and manipulating graphical representations of functions. These transformations, which can involve translations, reflections, stretches, and shears, provide valuable insights into the behavior and properties of functions. This article embarks on an exciting journey to uncover the rule that governs the transformation of the graph of the linear function y = 2x + 7 into the graph of y = 3x + 2. We will delve into the fundamental principles of graphical transformations, dissect the given functions, and employ a systematic approach to pinpoint the precise transformation rule. Prepare to witness the elegance and power of mathematical transformations as we unravel this intriguing problem.
Understanding Graphical Transformations
Before we embark on the quest to find the transformation rule, let's establish a firm understanding of the fundamental concepts underlying graphical transformations. These transformations, which act as the building blocks for manipulating graphs, can be broadly classified into several categories:
- Translations: These transformations involve shifting the graph either horizontally or vertically without altering its shape or size. A vertical translation shifts the graph up or down, while a horizontal translation shifts it left or right.
- Reflections: These transformations create a mirror image of the graph across a specific line, such as the x-axis or y-axis. Reflections across the x-axis change the sign of the y-coordinates, while reflections across the y-axis change the sign of the x-coordinates.
- Stretches and Compressions: These transformations alter the shape of the graph by either stretching it or compressing it along either the horizontal or vertical direction. Vertical stretches and compressions multiply the y-coordinates by a constant factor, while horizontal stretches and compressions multiply the x-coordinates by a constant factor.
- Shears: These transformations distort the graph by shifting points along a line parallel to one of the axes. Shears can be either horizontal or vertical, depending on the direction of the shift.
By understanding these basic transformations, we can effectively analyze and manipulate graphs of functions, gaining valuable insights into their behavior and properties.
Dissecting the Functions: y = 2x + 7 and y = 3x + 2
To successfully determine the transformation rule, we must first thoroughly analyze the given functions: y = 2x + 7 and y = 3x + 2. Both of these functions are linear functions, which means their graphs are straight lines. The general form of a linear function is y = mx + b, where m represents the slope and b represents the y-intercept.
Let's examine each function in detail:
- y = 2x + 7: This linear function has a slope of 2 and a y-intercept of 7. The slope of 2 indicates that for every 1 unit increase in x, the value of y increases by 2 units. The y-intercept of 7 indicates that the line intersects the y-axis at the point (0, 7).
- y = 3x + 2: This linear function has a slope of 3 and a y-intercept of 2. The slope of 3 indicates that for every 1 unit increase in x, the value of y increases by 3 units. The y-intercept of 2 indicates that the line intersects the y-axis at the point (0, 2).
By comparing the slopes and y-intercepts of these two functions, we can begin to discern the transformations that have occurred. The change in slope from 2 to 3 suggests a vertical stretch, while the change in y-intercept from 7 to 2 suggests a vertical translation.
Unveiling the Transformation Rule: A Step-by-Step Approach
Now, let's embark on the core of our investigation: determining the precise rule that transforms the graph of y = 2x + 7 into the graph of y = 3x + 2. We will employ a systematic approach, breaking down the transformation into a series of steps.
-
Vertical Stretch: The change in slope from 2 to 3 indicates a vertical stretch. To determine the stretch factor, we divide the new slope (3) by the original slope (2), which gives us a stretch factor of 3/2. This means the graph of y = 2x + 7 has been stretched vertically by a factor of 3/2.
Applying this vertical stretch, we obtain the intermediate function:
y = (3/2)(2x + 7) = 3x + 21/2
-
Vertical Translation: Next, we need to account for the change in y-intercept from 7 to 2. The y-intercept of the intermediate function y = 3x + 21/2 is 21/2, while the y-intercept of the target function y = 3x + 2 is 2. To achieve this change, we need to translate the graph vertically by a certain amount.
The vertical translation required is the difference between the new y-intercept (2) and the intermediate y-intercept (21/2), which is:
2 - 21/2 = -17/2
This indicates a vertical translation of -17/2 units, meaning the graph has been shifted downwards by 17/2 units.
Combining these two transformations, we arrive at the complete transformation rule: a vertical stretch by a factor of 3/2, followed by a vertical translation of -17/2 units.
In mathematical notation, this transformation can be expressed as:
y = (3/2)(2x + 7) - 17/2 = 3x + 2
Expressing the Transformation Rule: Function Notation
To provide a more concise and elegant representation of the transformation rule, we can utilize function notation. Let's denote the original function as f(x) = 2x + 7 and the transformed function as g(x) = 3x + 2. Our goal is to express g(x) in terms of f(x).
Based on our previous analysis, we know that the transformation involves a vertical stretch by a factor of 3/2 and a vertical translation of -17/2 units. We can express these operations using function notation as follows:
- Vertical Stretch: (3/2)f(x) = (3/2)(2x + 7) = 3x + 21/2
- Vertical Translation: (3/2)f(x) - 17/2 = (3x + 21/2) - 17/2 = 3x + 2
Therefore, we can express the transformed function g(x) in terms of the original function f(x) as:
g(x) = (3/2)f(x) - 17/2
This concise notation elegantly captures the transformation rule, highlighting the vertical stretch and vertical translation components.
Verification and Conclusion
To ensure the accuracy of our derived transformation rule, let's verify it by applying it to a specific point on the graph of y = 2x + 7 and confirming that the transformed point lies on the graph of y = 3x + 2.
Let's consider the point (0, 7) on the graph of y = 2x + 7. Applying our transformation rule, we first perform the vertical stretch by a factor of 3/2:
(3/2)(7) = 21/2
Then, we perform the vertical translation of -17/2 units:
21/2 - 17/2 = 4/2 = 2
Thus, the transformed point is (0, 2), which indeed lies on the graph of y = 3x + 2.
This verification strengthens our confidence in the derived transformation rule.
In conclusion, we have successfully unearthed the rule that transforms the graph of y = 2x + 7 into the graph of y = 3x + 2. This transformation involves a vertical stretch by a factor of 3/2, followed by a vertical translation of -17/2 units. By employing a systematic approach and leveraging the principles of graphical transformations, we have effectively solved this intriguing problem. The ability to identify and apply transformations is a fundamental skill in mathematics, empowering us to manipulate and understand the behavior of functions and their graphical representations.
Further Exploration: Generalizing Transformations
The principles we have employed in this article can be readily extended to more complex transformations and functions. Understanding how transformations affect the parameters of a function, such as the slope and y-intercept of a linear function, allows us to predict and manipulate the resulting graph. For instance, we can generalize our findings to explore the transformations that map one quadratic function to another, or even transformations involving trigonometric or exponential functions.
The world of transformations is vast and fascinating, offering endless opportunities for exploration and discovery. By mastering the fundamental concepts and techniques, we can unlock the power to manipulate and understand the intricate relationships between functions and their graphical representations.
Practice Problems
To solidify your understanding of transformations, try applying the techniques discussed in this article to the following practice problems:
- Find the rule for a transformation that takes the graph of y = x^2 to the graph of y = 2x^2 + 3.
- Determine the transformation that maps the graph of y = sin(x) to the graph of y = sin(x - π/2).
- Describe the sequence of transformations that takes the graph of y = e^x to the graph of y = -e^(x + 1).
By tackling these problems, you will hone your skills in identifying and applying transformations, further enhancing your mathematical prowess.
Conclusion
This article has delved into the captivating realm of graphical transformations, specifically focusing on the transformation that maps the graph of y = 2x + 7 to the graph of y = 3x + 2. We have meticulously dissected the functions, unveiled the transformation rule, and verified its accuracy. By understanding the fundamental principles of transformations, we can effectively manipulate and analyze graphs of functions, gaining valuable insights into their behavior and properties. The journey through transformations is an enriching experience, empowering us to appreciate the elegance and power of mathematics.
