Solving Exponential Equations Y = -3(4)^x For X = 2

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#Introduction

In the world of mathematics, exponential functions hold a place of importance, modeling diverse phenomena from population growth to radioactive decay. This article delves into a specific instance of such a function, focusing on the equation y=−3(4)xy = -3(4)^x and its solution when x=2x = 2. We'll dissect the function's characteristics, step through the solution process, and explore the broader implications of exponential functions in mathematical and real-world contexts. Understanding these functions is not only crucial for academic pursuits but also for grasping the underlying principles that govern many natural processes. This comprehensive exploration aims to illuminate the intricacies of exponential functions, providing a solid foundation for further mathematical endeavors. By meticulously examining the given equation, we can unravel the behavior of exponential relationships and appreciate their significance in various domains.

#Understanding Exponential Functions

At the heart of exponential functions lies a variable exponent, which dictates the rate of growth or decay. In the general form, y=a(b)xy = a(b)^x, the variable 'x' resides in the exponent, while 'a' represents the initial value and 'b' signifies the base, determining the rate of change. The base 'b' plays a pivotal role: if 'b' is greater than 1, the function exhibits exponential growth; conversely, if 'b' lies between 0 and 1, the function demonstrates exponential decay. The coefficient 'a' acts as a vertical stretch or compression factor, influencing the function's amplitude. Understanding these components is crucial for interpreting and manipulating exponential functions effectively. In our specific example, y=−3(4)xy = -3(4)^x, the base is 4, indicating exponential growth, while the coefficient -3 introduces a reflection across the x-axis and a vertical stretch. The negative sign also inverts the typical upward trajectory of an exponential growth function, causing it to decrease as x increases. This nuanced interplay of base and coefficient shapes the function's unique behavior and its graphical representation. Grasping these fundamental concepts allows us to predict and analyze the function's output for various input values, paving the way for solving more complex problems and applications involving exponential relationships. The elegance of exponential functions lies in their ability to capture dynamic changes with a concise mathematical expression, making them indispensable tools in diverse scientific and engineering fields.

#Solving the Equation y = -3(4)^x for x = 2

To solve the equation y=−3(4)xy = -3(4)^x for x=2x = 2, we embark on a straightforward substitution. Replacing 'x' with 2, we arrive at y=−3(4)2y = -3(4)^2. The next step involves evaluating the exponential term, 424^2, which equals 16. Subsequently, we multiply -3 by 16, yielding y=−48y = -48. This numerical solution provides a specific point on the graph of the exponential function, representing the y-value when x is 2. The calculation underscores the power of exponential growth, as the base 4, raised to the power of 2, significantly amplifies the initial coefficient of -3. The negative sign, however, inverts the result, positioning the point in the negative y-axis. This process exemplifies how exponential functions can produce dramatic changes in output even with relatively small changes in input. The simplicity of the substitution belies the profound implications of exponential relationships, which can lead to rapid increases or decreases depending on the base and exponent. This concrete example serves as a building block for understanding more complex scenarios involving exponential functions and their applications in various fields, from finance to population dynamics. The meticulous step-by-step solution highlights the importance of order of operations and the impact of each component of the function on the final result.

#Step-by-Step Solution

  1. Substitute x = 2 into the equation: y=−3(4)2y = -3(4)^2
  2. Evaluate the exponent: 42=164^2 = 16
  3. Multiply: y=−3∗16y = -3 * 16
  4. Calculate the final result: y=−48y = -48

Therefore, when x = 2, y = -48.

#Graphical Representation

The graphical representation of the function y=−3(4)xy = -3(4)^x offers valuable insights into its behavior. As x increases, the absolute value of y increases exponentially, but due to the negative coefficient, the graph is reflected across the x-axis. This means that instead of rising sharply as in a typical exponential growth function, the graph descends rapidly. The point (2, -48), which we calculated earlier, lies on this curve, providing a specific reference point. The graph never actually touches the x-axis (y = 0) because there is no value of x that can make 4x4^x equal to zero. This characteristic is known as having a horizontal asymptote at y = 0. The steep decline of the curve highlights the rapid change inherent in exponential functions, even with relatively small changes in x. The graph visually reinforces the concept of exponential growth, albeit in a downward direction due to the negative coefficient. Understanding the graphical representation of exponential functions is crucial for visualizing their behavior and predicting their outputs for various input values. It also aids in comparing and contrasting different exponential functions based on their bases and coefficients. The graph serves as a powerful tool for communicating the nature of exponential relationships and their implications in real-world scenarios.

#Implications and Applications

Exponential functions are not confined to the realm of pure mathematics; they permeate numerous real-world phenomena. Consider population growth, where the number of individuals increases exponentially over time, given sufficient resources. Radioactive decay, conversely, exemplifies exponential decay, as the amount of a radioactive substance diminishes exponentially. In finance, compound interest demonstrates exponential growth, where the accumulated interest earns further interest, leading to a rapid increase in investment value. The equation y=−3(4)xy = -3(4)^x, while a specific example, encapsulates the essence of exponential relationships, albeit with a negative coefficient that introduces a downward trend. This type of function might model scenarios where a quantity decreases exponentially, such as the depletion of a resource or the decay of a substance under certain conditions. The applications of exponential functions extend to diverse fields, including physics, chemistry, biology, economics, and computer science. Understanding these functions is therefore crucial for anyone seeking to model and analyze real-world phenomena accurately. The ability to manipulate and interpret exponential equations allows us to make predictions, optimize processes, and gain a deeper understanding of the dynamic world around us. From predicting the spread of diseases to designing efficient algorithms, exponential functions serve as indispensable tools for problem-solving and innovation.

#Conclusion

In conclusion, the equation y=−3(4)xy = -3(4)^x serves as a compelling illustration of exponential functions and their behavior. By substituting x=2x = 2, we determined that y=−48y = -48, a point that lies on the graph of this function. The negative coefficient introduces a reflection across the x-axis, resulting in a downward-sloping curve that demonstrates exponential decay in a negative direction. This example underscores the importance of understanding the components of an exponential function, including the base and the coefficient, and how they influence the function's behavior. Exponential functions, far from being abstract mathematical constructs, are fundamental tools for modeling real-world phenomena. Their applications span across various disciplines, from population dynamics and finance to physics and computer science. A solid grasp of exponential functions empowers us to analyze, predict, and optimize a wide range of processes and systems. This exploration of y=−3(4)xy = -3(4)^x provides a stepping stone for delving into more complex exponential models and their applications in diverse fields. The ability to interpret and manipulate these functions is essential for navigating the complexities of the modern world, where exponential growth and decay play a crucial role in shaping our understanding of natural and artificial systems.