Equivalent Equations For F(x) = -2(x-4) A Step-by-Step Guide
In the realm of mathematics, particularly when dealing with functions, it's crucial to understand how to identify equivalent equations. Equivalent equations represent the same relationship between variables, even if they appear different at first glance. This article delves into the process of determining which equation in two variables is equivalent to the given function, $f(x) = -2(x-4)$. We will explore various algebraic manipulations and comparisons to arrive at the correct answer, ensuring a clear understanding of the underlying principles.
Identifying the Correct Equivalent Equation
To find the equation that is equivalent to $f(x) = -2(x-4)$, we need to compare it with the provided options. The core task here involves simplifying the given function and comparing it with the other equations to see which one matches. Let's break down the function step by step. First, we distribute the -2 across the terms inside the parentheses:
This simplified form, $f(x) = -2x + 8$, is our benchmark. Now, let’s consider the options provided and manipulate them to see if they can be transformed into this form. The goal is to match the equation's slope and y-intercept to confirm equivalence. Understanding these steps is vital for high school mathematics students who are studying linear functions and their representations. Equivalent equations are not just a theoretical concept; they have practical applications in solving complex problems and simplifying mathematical models.
Analyzing the First Option: y = -2(f(x) - 4)
The first option given is $y = -2(f(x) - 4)$. This equation introduces a slightly more complex scenario where the function f(x) is part of the equation itself. To determine if this equation is equivalent, we need to substitute the expression for f(x) that we initially simplified, which is $f(x) = -2x + 8$. By substituting and further simplifying, we can assess whether it aligns with our original function. Let's perform the substitution:
Upon simplification, we obtain $y = 4x - 8$. Comparing this to our benchmark simplified function, $f(x) = -2x + 8$, it’s evident that these two equations are not the same. The slopes and y-intercepts differ significantly. Therefore, the equation $y = -2(f(x) - 4)$ is not equivalent to the original function. This process highlights the importance of meticulous algebraic manipulation and comparison when dealing with mathematical equations. Recognizing non-equivalent forms is as crucial as identifying equivalent ones.
Evaluating the Second Option: y = -2x + 4
The second option is $y = -2x + 4$. This equation appears simpler than the first, but we still need to verify its equivalence to the given function, $f(x) = -2x + 8$. Directly comparing this equation with our simplified function, we can observe that the slopes are the same (-2), but the y-intercepts are different. The given function has a y-intercept of 8, while this option has a y-intercept of 4. Since the y-intercepts do not match, the equation $y = -2x + 4$ is not equivalent to the original function. This illustrates a key concept in mathematics: for two linear equations to be equivalent, both their slopes and y-intercepts must be identical. Any variation in these parameters leads to a different graphical representation and, therefore, a non-equivalent equation.
Assessing the Third Option: y = -2(f(x) + 4)
The third option presented is $y = -2(f(x) + 4)$. Similar to the first option, this equation involves substituting the expression for f(x). We will replace f(x) with its simplified form, $f(x) = -2x + 8$, and proceed with algebraic simplification to determine if the resulting equation is equivalent to our benchmark. This involves careful distribution and combining like terms to reveal the true form of the equation. Let's perform the substitution and simplification:
After simplifying, we obtain the equation $y = 4x - 24$. Comparing this with our original function, $f(x) = -2x + 8$, it is clear that they are not equivalent. The slopes are different (4 versus -2), and the y-intercepts also differ (-24 versus 8). This further reinforces the principle that equivalent linear equations must have the same slope and y-intercept. This exercise is essential for students to grasp the nuances of mathematical equivalence and the importance of accurate algebraic manipulation.
Determining Equivalence for the Fourth Option: y = -2(x - 4)
The final option to consider is $y = -2(x - 4)$. This equation closely resembles the original function, $f(x) = -2(x - 4)$, but to confirm its equivalence, we still need to simplify and compare it. This step is crucial to ensure there are no hidden differences. We distribute the -2 across the terms inside the parentheses, as we did initially, to reveal the equation's simplified form. This process will allow us to directly compare it with our benchmark simplified equation and definitively determine if they are equivalent. Let's simplify the equation:
Upon simplification, we arrive at $y = -2x + 8$. Comparing this to our simplified function, $f(x) = -2x + 8$, we can see that they are exactly the same. Both the slopes and y-intercepts match perfectly. Therefore, the equation $y = -2(x - 4)$ is indeed equivalent to the original function. This confirms the importance of methodical simplification and comparison in mathematical problem-solving. The ability to recognize equivalent forms is fundamental for advanced mathematical concepts and applications.
Conclusion: The Equivalent Equation
After carefully analyzing each option, we can definitively conclude that the equation equivalent to the function $f(x) = -2(x - 4)$ is $y = -2(x - 4)$. This determination was made through a process of algebraic simplification and direct comparison. The key steps involved distributing the constants, identifying the slopes and y-intercepts, and matching them across equations. This exercise underscores the significance of precision in mathematical manipulations and the necessity of understanding the underlying principles of equation equivalence. The correct identification of equivalent equations is a cornerstone of advanced mathematical studies, enabling students to solve complex problems and build a solid foundation in algebraic concepts. The process also highlights how seemingly different equations can represent the same relationship between variables, which is a critical concept in various fields, including mathematics, physics, and engineering.
