Equivalent Equation For F(x) = -2(x - 4) A Step-by-Step Solution

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In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships between variables. One common task involves expressing a function in different forms while preserving its underlying meaning. This article delves into the process of identifying an equation in two variables that mirrors the behavior of the function f(x) = -2(x - 4). We will dissect the given function, explore potential equivalent equations, and ultimately pinpoint the correct answer through a step-by-step analysis.

Understanding the Function f(x) = -2(x - 4)

At its core, the function f(x) = -2(x - 4) defines a relationship between an input variable x and an output value f(x). The function operates by first subtracting 4 from the input x, and then multiplying the result by -2. To gain a deeper understanding of this function, let's break down its components:

  1. Input Variable (x): The variable x represents the independent variable, which can take on various values.
  2. Subtraction (x - 4): This operation shifts the graph of the function 4 units to the right along the x-axis.
  3. Multiplication by -2 (-2(x - 4)): Multiplying by -2 performs two transformations:
    • Vertical Stretch: The graph is stretched vertically by a factor of 2.
    • Reflection across the x-axis: The graph is flipped over the x-axis due to the negative sign.
  4. Output Value (f(x)): The result of applying the operations to x is the output value f(x), which represents the dependent variable.

To further solidify our grasp, let's evaluate the function for a few sample values of x:

  • If x = 0, then f(0) = -2(0 - 4) = -2(-4) = 8
  • If x = 2, then f(2) = -2(2 - 4) = -2(-2) = 4
  • If x = 4, then f(4) = -2(4 - 4) = -2(0) = 0
  • If x = 6, then f(6) = -2(6 - 4) = -2(2) = -4

These evaluations provide concrete examples of how the function transforms input values into output values. By understanding the individual operations and their effects, we can better analyze potential equivalent equations.

Exploring Potential Equivalent Equations

The challenge lies in expressing the function f(x) = -2(x - 4) as an equation in two variables. This means we need to introduce a second variable, typically denoted as y, and establish a relationship between x and y that mirrors the behavior of the original function. Let's examine the provided options and dissect their potential equivalence:

Option 1: y = -2(f(x) - 4)

This equation presents a seemingly complex relationship between y and f(x). To unravel its meaning, let's substitute the expression for f(x) from the original function:

y = -2((-2(x - 4)) - 4)

Now, we can simplify the expression:

y = -2(-2x + 8 - 4) y = -2(-2x + 4) y = 4x - 8

This equation represents a linear relationship between x and y with a slope of 4 and a y-intercept of -8. However, it's crucial to compare this equation with the direct translation of the original function into two variables.

Option 2: y = -2x + 4

This equation represents a linear relationship between x and y with a slope of -2 and a y-intercept of 4. To determine if this equation is equivalent, we need to directly translate the original function into two variables.

Option 3: y = -2(f(x) + 4)

Similar to the first option, this equation involves f(x) within its expression. Let's substitute the expression for f(x):

y = -2((-2(x - 4)) + 4)

Simplifying the expression:

y = -2(-2x + 8 + 4) y = -2(-2x + 12) y = 4x - 24

This equation also represents a linear relationship, but with a slope of 4 and a y-intercept of -24. Again, we need to compare this with the direct translation.

Option 4: y = -2(x - 4)

This equation appears to be the most direct translation of the original function into two variables. To confirm, let's analyze it further.

Pinpointing the Correct Answer: A Step-by-Step Analysis

To identify the correct equation, we need to directly translate the function f(x) = -2(x - 4) into an equation in two variables. The most natural way to do this is to replace f(x) with y, representing the output value:

y = -2(x - 4)

This equation directly mirrors the operations defined in the original function. It subtracts 4 from x and then multiplies the result by -2, just like f(x). Therefore, this equation is the most likely candidate for equivalence.

To further validate our answer, let's simplify the equation and compare it with the simplified forms of the other options:

y = -2(x - 4) y = -2x + 8

This simplified form represents a linear equation with a slope of -2 and a y-intercept of 8. Now, let's compare this with the simplified forms of the other options:

  • Option 1: y = 4x - 8 (Slope: 4, y-intercept: -8)
  • Option 2: y = -2x + 4 (Slope: -2, y-intercept: 4)
  • Option 3: y = 4x - 24 (Slope: 4, y-intercept: -24)
  • Option 4: y = -2x + 8 (Slope: -2, y-intercept: 8)

By comparing the slopes and y-intercepts, we can definitively conclude that Option 4 (y = -2(x - 4)) is the only equation that is truly equivalent to the original function f(x) = -2(x - 4). It shares the same slope and y-intercept, indicating that it represents the same linear relationship between x and y.

Conclusion: The Power of Direct Translation

In this exploration, we successfully identified the equation in two variables that is equivalent to the function f(x) = -2(x - 4). The key to solving this problem lay in the understanding of the function's operations and the direct translation into an equation in two variables. By replacing f(x) with y, we created an equation that faithfully mirrored the relationship between the input x and the output.

We also demonstrated the importance of simplifying equations and comparing their key features, such as slope and y-intercept, to validate our answer. Through this process, we not only found the correct equation but also gained a deeper appreciation for the interconnectedness of different representations of mathematical functions. This understanding is crucial for tackling more complex mathematical problems and building a strong foundation in algebra and beyond.

This exercise highlights the fundamental principle that equivalent equations represent the same mathematical relationship, even if they appear different at first glance. By carefully analyzing the operations and transformations involved, we can confidently navigate the world of functions and equations, unlocking their power to model and understand the world around us.