Identifying Exponential Functions Passing Through A Point (2, 36)

An exponential function is a function where the independent variable (x) is in the exponent. These functions are characterized by their rapid growth or decay. Identifying the correct exponential function that passes through a given point requires understanding the general form of an exponential equation and how to use the given point to validate the equation.

Understanding Exponential Functions

Before diving into the problem, let's clarify the basic structure of an exponential function. The general form is typically expressed as:

f(x) = a * b^x

Where:

• f(x) is the value of the function at x.
• a is the initial value or the y-intercept (the value of f(x) when x = 0).
• b is the base, which determines the rate of growth (if b > 1) or decay (if 0 < b < 1).
• x is the independent variable.

Key Characteristics of Exponential Functions

• Exponential functions do not have a constant rate of change; instead, they have a constant ratio of change. This means that for every unit increase in x, the function value is multiplied by a constant factor (the base, b).
• The graph of an exponential function either increases or decreases rapidly, depending on whether the base b is greater than 1 or between 0 and 1, respectively.
• Exponential functions are used to model various real-world phenomena, including population growth, radioactive decay, and compound interest.

Problem Breakdown

The question asks us to identify the exponential function that passes through the point (2, 36). This means that when x = 2, f(x) = 36. We are given four options:

A. f(x) = 4(3)^x B. f(x) = 4(x)^3 C. f(x) = 6(3)^x D. f(x) = 6(x)^3

To solve this, we need to substitute x = 2 into each equation and see which one results in f(x) = 36. This process will help us validate which of the provided equations correctly models the exponential relationship passing through the given point.

Step-by-Step Solution

Let's analyze each option:

Option A: f(x) = 4(3)^x

Substitute x = 2:

f(2) = 4(3)^2 f(2) = 4 * 9 f(2) = 36

This equation holds true for the point (2, 36).

Option B: f(x) = 4(x)^3

Substitute x = 2:

f(2) = 4(2)^3 f(2) = 4 * 8 f(2) = 32

This equation does not pass through the point (2, 36).

Option C: f(x) = 6(3)^x

Substitute x = 2:

f(2) = 6(3)^2 f(2) = 6 * 9 f(2) = 54

This equation does not pass through the point (2, 36).

Option D: f(x) = 6(x)^3

Substitute x = 2:

f(2) = 6(2)^3 f(2) = 6 * 8 f(2) = 48

This equation also does not pass through the point (2, 36).

Conclusion

After evaluating all options, only equation A, f(x) = 4(3)^x, satisfies the condition of passing through the point (2, 36). This makes option A the correct answer.

Deep Dive into Exponential Function Properties

To further solidify understanding, let's explore some deeper properties of exponential functions. Understanding these properties will not only help in solving similar problems but also in applying exponential functions to real-world scenarios.

The Base (b) and Its Implications

The base b in the exponential function f(x) = a * b^x plays a critical role in determining the behavior of the function.

• If b > 1, the function represents exponential growth. As x increases, f(x) increases rapidly. Larger values of b indicate faster growth rates. For instance, comparing f(x) = 2^x and f(x) = 3^x, the latter grows much faster because its base is larger.
• If 0 < b < 1, the function represents exponential decay. As x increases, f(x) decreases rapidly, approaching zero. The closer b is to 0, the faster the decay. An example is the decay of radioactive substances, where the amount of substance remaining decreases exponentially over time.
• If b = 1, the function becomes f(x) = a * 1^x = a, which is a constant function, not an exponential function.
• If b ≤ 0, the function is not well-defined for all real numbers, especially when x is not an integer.

The Initial Value (a)

The initial value a in f(x) = a * b^x represents the value of the function when x = 0. It is the y-intercept of the graph of the exponential function. The initial value scales the exponential function. If a is positive, the function's values are positive, and if a is negative, the function's values are negative.

Transformations of Exponential Functions

Understanding transformations can help in visualizing and manipulating exponential functions. Common transformations include:

• Vertical Shifts: Adding a constant to the function, f(x) + c, shifts the graph vertically. If c > 0, the graph shifts upward; if c < 0, it shifts downward.
• Horizontal Shifts: Replacing x with (x - h) in the function, f(x - h), shifts the graph horizontally. If h > 0, the graph shifts to the right; if h < 0, it shifts to the left.
• Vertical Stretches and Compressions: Multiplying the function by a constant, c * f(x), stretches the graph vertically if c > 1 and compresses it if 0 < c < 1. If c < 0, it also reflects the graph across the x-axis.
• Horizontal Stretches and Compressions: Replacing x with cx in the function, f(cx), compresses the graph horizontally if c > 1 and stretches it if 0 < c < 1.

Real-World Applications

Exponential functions are pivotal in modeling numerous real-world scenarios. Here are a few prominent examples:

1. Population Growth: Exponential growth models are used to describe how populations increase over time when resources are abundant. The basic model assumes that the growth rate is proportional to the current population size. The equation often used is P(t) = P₀ * e^(kt), where P(t) is the population at time t, P₀ is the initial population, e is the base of the natural logarithm (approximately 2.71828), and k is the growth constant.
2. Compound Interest: Compound interest is a classic example of exponential growth. The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial deposit), r is the annual interest rate (as a decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for, in years.
3. Radioactive Decay: Radioactive decay follows an exponential decay model. The amount of a radioactive substance decreases exponentially over time. The half-life of a radioactive isotope is the time it takes for half of the substance to decay. The formula for radioactive decay is N(t) = N₀ * e^(-λt), where N(t) is the amount of the substance remaining at time t, N₀ is the initial amount of the substance, e is the base of the natural logarithm, and λ (lambda) is the decay constant, which is related to the half-life of the substance.
4. Spread of Diseases: The spread of infectious diseases can often be modeled using exponential functions, especially in the early stages of an outbreak. The number of infected individuals can grow exponentially before measures are taken to control the spread. Epidemiological models, such as the SIR (Susceptible-Infected-Recovered) model, incorporate exponential growth and decay to predict the progression of a disease.
5. Learning Curves: In psychology and organizational behavior, learning curves often exhibit an exponential pattern. The rate at which a person learns a new skill or task typically decreases over time. Initially, there is rapid improvement, but as proficiency increases, the rate of learning slows down. Exponential functions can be used to model this phenomenon, helping to understand and predict learning rates.

Common Mistakes and How to Avoid Them

When dealing with exponential functions, there are several common mistakes students often make. Recognizing these pitfalls can help you avoid errors and improve your problem-solving skills.

1. Confusing Exponential and Polynomial Functions: One of the most common mistakes is confusing exponential functions with polynomial functions. An exponential function has the variable in the exponent (e.g., f(x) = 2^x), whereas a polynomial function has the variable as the base (e.g., f(x) = x^2). Always check where the variable is located to correctly identify the type of function.
2. Incorrectly Applying the Order of Operations: When evaluating an exponential function, it is crucial to follow the order of operations (PEMDAS/BODMAS). Exponents should be calculated before multiplication. For example, in the function f(x) = 3 * 2^x, you must calculate 2^x first and then multiply the result by 3.
3. Misinterpreting the Base: The base of an exponential function determines whether the function represents growth or decay. If the base is greater than 1, it’s growth; if it’s between 0 and 1, it’s decay. Misinterpreting the base can lead to incorrect predictions and analyses.
4. Forgetting the Initial Value: The initial value (a) in f(x) = a * b^x is the y-intercept of the function and plays a crucial role in scaling the function. Forgetting to include or incorrectly calculating the initial value can lead to significant errors.
5. Errors with Negative Exponents: Negative exponents indicate reciprocals. For example, 2^(-x) is the same as 1 / (2^x). Mistakes in handling negative exponents are common, so it's essential to remember the rules of exponents.
6. Incorrectly Substituting Values: When solving problems that involve finding a specific exponential function that passes through a given point, ensure that you substitute the x and y values correctly. Recheck your substitutions to avoid simple arithmetic errors.
7. Ignoring Transformations: Transformations such as shifts and stretches can significantly alter the graph and equation of an exponential function. Failing to account for these transformations can result in an incorrect analysis. Always consider how transformations affect the base function.

Tips to Avoid Mistakes

• Practice Regularly: Consistent practice is key to mastering exponential functions. Work through a variety of problems to reinforce your understanding.
• Double-Check Calculations: Always double-check your calculations, especially when dealing with exponents and substitutions.
• Understand the Concepts: Focus on understanding the fundamental concepts rather than memorizing formulas. A strong conceptual foundation will help you apply the knowledge correctly.
• Graph the Functions: Visualizing exponential functions through graphs can help you understand their behavior and identify potential errors.
• Use Real-World Examples: Relate exponential functions to real-world scenarios to make the concepts more concrete and easier to remember.

Final Thoughts

Understanding exponential functions is essential for various mathematical and real-world applications. By grasping the core concepts, properties, and common pitfalls, you can confidently tackle problems involving exponential growth and decay. Remember to practice consistently, double-check your work, and visualize the functions to enhance your comprehension. Whether it’s determining population growth, calculating compound interest, or understanding radioactive decay, a solid foundation in exponential functions is invaluable.

By systematically substituting the given point into each equation, we identified that option A, f(x) = 4(3)^x, is the only function that passes through (2, 36). This exercise underscores the importance of understanding exponential functions and their properties in solving mathematical problems and applying them to real-world scenarios.