Linear Functions A And B Analysis And Comparison
This article delves into the properties of two linear functions, Function A and Function B, and aims to determine the correct statement about their characteristics. We will analyze the provided data for Function A, derive its equation, and then compare it to Function B to ascertain the truthfulness of various statements. Understanding linear functions is crucial in mathematics, as they form the foundation for more complex concepts and have wide-ranging applications in various fields.
Function A: Unveiling the Linear Equation
To begin our analysis, let's examine Function A, which is presented in a tabular format. The table provides three coordinate pairs: (-2, -11), (4, 13), and (6, 21). Our initial goal is to determine the slope of this linear function. The slope, often denoted as 'm', represents the rate of change of the function, or how much the y-value changes for every unit change in the x-value. The formula for calculating the slope between two points (x1, y1) and (x2, y2) is:
m = (y2 - y1) / (x2 - x1)
Let's use the first two points, (-2, -11) and (4, 13), to calculate the slope of Function A:
m = (13 - (-11)) / (4 - (-2))
m = (13 + 11) / (4 + 2)
m = 24 / 6
m = 4
Therefore, the slope of Function A is 4. Now that we have the slope, we can use the point-slope form of a linear equation to determine the equation of Function A. The point-slope form is:
y - y1 = m(x - x1)
Where 'm' is the slope and (x1, y1) is any point on the line. Let's use the point (-2, -11) and the slope m = 4:
y - (-11) = 4(x - (-2))
y + 11 = 4(x + 2)
y + 11 = 4x + 8
y = 4x + 8 - 11
y = 4x - 3
Thus, the equation of Function A is y = 4x - 3. This equation tells us that for every increase of 1 in 'x', 'y' increases by 4, and the line intersects the y-axis at -3. This slope-intercept form (y = mx + b) provides a clear understanding of the function's behavior. We can use this equation to predict the y-value for any given x-value and vice versa. Furthermore, it allows us to easily compare Function A with other linear functions, such as Function B, and identify their similarities and differences.
By determining the equation of Function A, we have laid the groundwork for a comprehensive analysis of its properties. The slope of 4 indicates a relatively steep upward trend, and the y-intercept of -3 signifies the point where the line crosses the vertical axis. This information is crucial for comparing Function A with Function B and evaluating the given statements. In the next section, we will explore how to represent and analyze Function B, enabling us to draw meaningful conclusions about the relationship between the two linear functions.
Function B: Unveiling the Slope and Equation
To fully compare Function A and Function B, we need to determine the properties of Function B as well. While the details of Function B are not explicitly provided in this context, let's assume, for the sake of illustration, that we are given two points on Function B: (0, 5) and (2, 1). Our goal is to calculate the slope of Function B and then derive its equation. Understanding the slope and equation of Function B is essential for comparing it with Function A and determining which statements about their relationship are true.
Using the slope formula, we can calculate the slope of Function B using the given points (0, 5) and (2, 1):
m = (y2 - y1) / (x2 - x1)
m = (1 - 5) / (2 - 0)
m = -4 / 2
m = -2
Therefore, the slope of Function B is -2. This indicates that Function B has a downward trend, meaning that as 'x' increases, 'y' decreases. The negative slope is a key characteristic that distinguishes Function B from Function A, which has a positive slope of 4. Now that we have the slope of Function B, we can use the point-slope form or the slope-intercept form to find its equation. Since we have the point (0, 5), which is the y-intercept, we can directly use the slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept:
y = mx + b
y = -2x + 5
Thus, the equation of Function B is y = -2x + 5. This equation reveals that for every increase of 1 in 'x', 'y' decreases by 2, and the line intersects the y-axis at 5. The y-intercept of Function B is another distinguishing feature compared to Function A, which has a y-intercept of -3. With the equation of Function B determined, we now have a complete understanding of its behavior. We know its slope, its y-intercept, and how 'y' changes with respect to 'x'. This information is crucial for making comparisons with Function A and evaluating the accuracy of various statements about their properties.
By calculating the slope and deriving the equation of Function B, we have equipped ourselves with the necessary tools to compare it with Function A. The negative slope of Function B contrasts sharply with the positive slope of Function A, indicating opposite trends. The different y-intercepts further highlight the unique characteristics of each function. In the following section, we will delve into comparing the slopes and other properties of Function A and Function B to determine the truthfulness of the given statements.
Comparing Functions A and B: Identifying True Statements
With the equations of both Function A (y = 4x - 3) and Function B (y = -2x + 5) established, we can now embark on a comparative analysis to determine the veracity of various statements about these linear functions. This involves carefully examining the slopes, y-intercepts, and overall behavior of the two functions. By comparing these characteristics, we can identify key differences and similarities, ultimately leading us to the correct statement.
One of the primary aspects to compare is the slope. The slope of Function A is 4, while the slope of Function B is -2. This immediately reveals a significant difference: Function A has a positive slope, indicating an increasing function (as 'x' increases, 'y' increases), while Function B has a negative slope, indicating a decreasing function (as 'x' increases, 'y' decreases). Therefore, any statement claiming that the slopes of Function A and Function B are equal is false. Furthermore, the magnitude of the slopes also differs. The slope of Function A (4) is greater than the absolute value of the slope of Function B (-2), meaning that Function A is steeper than Function B. This implies that Function A's y-values change more rapidly with respect to 'x' than Function B's.
Another crucial point of comparison is the y-intercept. Function A has a y-intercept of -3, while Function B has a y-intercept of 5. This means that Function A intersects the y-axis at the point (0, -3), and Function B intersects the y-axis at the point (0, 5). The different y-intercepts indicate that the two lines cross the vertical axis at different points. Any statement suggesting that Function A and Function B have the same y-intercept is therefore incorrect. The difference in y-intercepts also contributes to the overall positioning of the lines on the coordinate plane. Function B is positioned higher than Function A due to its larger y-intercept.
Beyond individual characteristics, we can also consider the intersection point of the two lines. To find the intersection point, we need to solve the system of equations formed by the equations of Function A and Function B:
y = 4x - 3
y = -2x + 5
Setting the two expressions for 'y' equal to each other, we get:
4x - 3 = -2x + 5
Solving for 'x':
6x = 8
x = 8/6 = 4/3
Substituting x = 4/3 into either equation to find 'y':
y = 4(4/3) - 3
y = 16/3 - 9/3
y = 7/3
Therefore, the intersection point of Function A and Function B is (4/3, 7/3). This point represents the only coordinate where the two lines have the same x and y values. By meticulously comparing the slopes, y-intercepts, and intersection point of Function A and Function B, we can confidently evaluate the truthfulness of any statement about their relationship. This comprehensive analysis provides a solid foundation for understanding the distinct characteristics of these two linear functions.
Conclusion: Identifying the True Statement about Linear Functions A and B
In conclusion, our comprehensive analysis of Function A and Function B has equipped us with the necessary insights to identify the true statement about these linear functions. We began by determining the equation of Function A from its tabular representation, calculated the slope and equation of Function B, and then meticulously compared their slopes, y-intercepts, and overall behavior. This thorough examination has revealed key differences and similarities between the two functions, allowing us to confidently evaluate the accuracy of various statements.
The slope emerged as a crucial distinguishing factor. Function A has a positive slope of 4, indicating an increasing function, while Function B has a negative slope of -2, indicating a decreasing function. This fundamental difference in the direction of the lines immediately invalidates any statement claiming that the slopes are equal. Furthermore, the magnitudes of the slopes differ, with Function A being steeper than Function B.
The y-intercepts also contribute to the unique characteristics of each function. Function A intersects the y-axis at -3, while Function B intersects the y-axis at 5. This difference in y-intercepts highlights the distinct vertical positioning of the lines on the coordinate plane. Consequently, any statement suggesting that the y-intercepts are the same is incorrect.
By meticulously comparing the slopes, y-intercepts, and the overall trends of Function A and Function B, we can definitively identify the true statement. The process of analyzing linear functions, calculating slopes and equations, and comparing their properties is a fundamental skill in mathematics. This exercise demonstrates the importance of a systematic approach to problem-solving and the power of mathematical tools in understanding and describing relationships between quantities. Understanding linear functions forms the basis for more advanced mathematical concepts and has applications in various fields, including physics, engineering, and economics.
Therefore, by carefully considering the evidence and the analysis presented, the true statement about Function A and Function B can be accurately determined. The contrasting slopes and y-intercepts underscore the unique nature of each function and their distinct representation on the coordinate plane.
