Finding Exponential Functions An Equation That Passes Through (2, 36)

To determine which equation represents an exponential function that passes through the point $(2, 36)$, we need to delve into the fundamental characteristics of exponential functions and how to verify if a given point lies on the graph of a function. Exponential functions are a cornerstone of mathematics, particularly in calculus and its applications, because they efficiently model many real-world phenomena, including population growth, compound interest, and radioactive decay. An exponential function is defined by the general form $f(x) = a imes b^x$, where $a$ is the initial value (the y-intercept), $b$ is the base (a positive real number not equal to 1), and $x$ is the exponent. The point $(2, 36)$ means that when $x = 2$, the function value $f(x)$ must equal 36. To find the correct equation, we substitute $x = 2$ into each given function and check which one yields $f(2) = 36$. This process involves careful arithmetic and a clear understanding of the order of operations. It is also essential to distinguish between exponential functions and polynomial functions, as they behave very differently. An exponential function has a constant base raised to a variable exponent, while a polynomial function has a variable base raised to a constant exponent. For example, $f(x) = 2^x$ is an exponential function, whereas $f(x) = x^2$ is a polynomial function. The rate of growth of an exponential function is much faster than that of a polynomial function as $x$ becomes large. This rapid growth is a hallmark of exponential behavior and is critical in modeling phenomena that exhibit similar patterns, such as the spread of a disease or the accumulation of money in a savings account with compound interest. Understanding the properties of exponential functions is crucial for solving a wide range of mathematical problems and for applying these concepts in practical scenarios.

Evaluating the Options

To identify the exponential function that passes through the point $(2, 36)$, we need to systematically evaluate each option by substituting $x = 2$ into the function and checking if the result is equal to 36. This is a straightforward application of the definition of a function and the meaning of a point lying on a function's graph. First, let's examine option A, $f(x) = 4(3)^x$. Substituting $x = 2$, we get $f(2) = 4(3)^2 = 4(9) = 36$. This result matches the given y-value of 36, making option A a potential candidate. Next, we evaluate option B, $f(x) = 4(x)^3$. Substituting $x = 2$, we get $f(2) = 4(2)^3 = 4(8) = 32$. This does not equal 36, so option B is not the correct answer. Option B represents a cubic function, not an exponential function, and serves as a good example of the distinction between polynomial and exponential functions. The behavior of a cubic function is markedly different from that of an exponential function; cubic functions increase at a polynomial rate, whereas exponential functions increase at an exponential rate. Then, let's consider option C, $f(x) = 6(3)^x$. Substituting $x = 2$, we get $f(2) = 6(3)^2 = 6(9) = 54$. This is not equal to 36, so option C is also incorrect. The initial coefficient of 6 instead of 4 leads to a different scaling of the exponential term, resulting in a higher value at $x = 2$. Finally, we evaluate option D, $f(x) = 6(x)^3$. Substituting $x = 2$, we get $f(2) = 6(2)^3 = 6(8) = 48$. This is also not equal to 36, so option D is not the correct answer. Like option B, option D is a cubic function and not an exponential function, further highlighting the importance of recognizing the form of exponential functions. Through this step-by-step evaluation, we can methodically determine which function, if any, satisfies the given condition. This process underscores the importance of careful substitution and accurate calculation in mathematical problem-solving.

The Correct Equation

Based on our evaluations, only option A, $f(x) = 4(3)^x$, satisfies the condition of passing through the point $(2, 36)$. By substituting $x = 2$ into the function, we obtained $f(2) = 4(3)^2 = 4(9) = 36$, which confirms that the point lies on the graph of this function. This result highlights the significance of the initial coefficient and the base in an exponential function. The coefficient of 4 scales the exponential term, while the base of 3 determines the rate of exponential growth. A different coefficient or base would result in a different function value at $x = 2$, and thus, the point $(2, 36)$ would not lie on the graph. The other options, B, C, and D, did not yield 36 when $x = 2$, eliminating them as possibilities. Options B and D were cubic functions, illustrating that the form of the function is crucial in determining its behavior and the points it passes through. Option C was an exponential function but with a different coefficient, demonstrating how even a slight change in the function's parameters can significantly alter its output. The fact that only one option matches the condition reinforces the importance of precision and careful evaluation in mathematics. This exercise not only identifies the correct equation but also deepens our understanding of how exponential functions work and how they differ from other types of functions. Understanding the role of each parameter in an exponential function is essential for modeling various real-world phenomena and for solving related mathematical problems. Therefore, the correct equation is $f(x) = 4(3)^x$, which accurately represents an exponential function passing through the point $(2, 36)$.

Why Other Options Are Incorrect

Understanding why the other options are incorrect is just as important as identifying the correct answer. It reinforces the fundamental principles of exponential functions and helps to avoid common mistakes. Option B, $f(x) = 4(x)^3$, is a cubic function, not an exponential function. In a cubic function, the variable $x$ is raised to a constant power (in this case, 3), whereas, in an exponential function, a constant base is raised to a variable power. When we substitute $x = 2$ into this equation, we get $f(2) = 4(2)^3 = 4(8) = 32$, which is not equal to 36. This demonstrates that the point $(2, 36)$ does not lie on the graph of this cubic function. This distinction between exponential and polynomial functions is crucial in mathematics, as they exhibit very different behaviors and are used to model different types of phenomena. Option B’s error highlights the need to recognize the fundamental form of exponential functions and differentiate them from other types of functions.

Option C, $f(x) = 6(3)^x$, is an exponential function, but the coefficient is different from the correct answer. Substituting $x = 2$ into this equation yields $f(2) = 6(3)^2 = 6(9) = 54$, which is also not equal to 36. This shows how a change in the coefficient can affect the function's value at a particular point. While the base of 3 indicates exponential growth, the coefficient of 6 scales the function differently compared to the correct equation with a coefficient of 4. This discrepancy underscores the importance of both the base and the coefficient in determining the function's behavior and the points it passes through. A clear understanding of how coefficients and bases affect exponential functions is essential for accurate modeling and problem-solving.

Option D, $f(x) = 6(x)^3$, like option B, is a cubic function, not an exponential function. Substituting $x = 2$ into this equation gives us $f(2) = 6(2)^3 = 6(8) = 48$, which is not equal to 36. Again, this highlights the crucial distinction between exponential and polynomial functions. The variable $x$ being raised to a constant power instead of a constant base being raised to a variable power is the defining characteristic of a polynomial function, and it leads to a different growth pattern compared to exponential functions. Recognizing the form of an equation is crucial for identifying the type of function it represents and for applying the appropriate mathematical techniques to analyze it. In summary, the incorrect options serve as valuable learning opportunities, reinforcing the properties of exponential functions and the importance of accurate evaluation and attention to detail.

Conclusion

In conclusion, determining which equation represents an exponential function that passes through a given point requires a careful understanding of the function's form and systematic evaluation. The correct equation, $f(x) = 4(3)^x$, satisfies the condition of passing through the point $(2, 36)$, as demonstrated by substituting $x = 2$ and confirming that $f(2) = 36$. This exercise underscores the importance of recognizing the general form of an exponential function, $f(x) = a imes b^x$, and the roles of the coefficient $a$ and the base $b$ in shaping the function's behavior. The coefficient scales the exponential term, while the base determines the rate of growth. By evaluating each option, we methodically eliminated those that did not satisfy the condition, reinforcing the need for precision in mathematical calculations. Options B and D were cubic functions, highlighting the distinction between polynomial and exponential functions. Option C was an exponential function with a different coefficient, demonstrating how even a slight change in the parameters can significantly alter the function's output. Through this process, we not only identified the correct equation but also deepened our understanding of exponential functions and their properties. This comprehensive approach is essential for solving a wide range of mathematical problems and for applying these concepts in real-world scenarios. By understanding why the incorrect options fail to meet the given condition, we strengthen our grasp of the underlying mathematical principles. This knowledge enables us to approach similar problems with confidence and accuracy. Ultimately, this exercise reinforces the importance of both conceptual understanding and meticulous application of mathematical techniques in problem-solving.