How To Complete The Table Of Values For Exponential Functions

Hey guys! Today, we're diving into completing a table of values for an exponential function. This is a crucial skill in mathematics, especially when dealing with exponential growth and decay. We'll break down the process step-by-step, making it super easy to understand. So, let's get started!

Understanding Exponential Functions

Before we jump into the table, let's quickly recap what an exponential function is. An exponential function generally looks like this: f(x) = a^x, where 'a' is a constant base and 'x' is the exponent. The base 'a' can be any positive number except 1. These functions are essential in modeling various real-world phenomena, such as population growth, compound interest, and radioactive decay. Exponential functions are characterized by their rapid growth or decay as the variable 'x' changes. Understanding their behavior is key to solving many mathematical problems and real-world applications.

When working with exponential functions, it's crucial to remember the basic rules of exponents. For instance, any number raised to the power of 0 is 1, and a number raised to a negative power is the reciprocal of the number raised to the positive power. These rules help simplify calculations and make it easier to fill in the table of values. By mastering these concepts, you'll be able to tackle more complex problems involving exponential functions and their applications in various fields. Furthermore, recognizing the properties of exponential functions, such as their increasing or decreasing nature based on the base 'a', can provide valuable insights into the behavior of the function.

The Given Function: (2/3)^x

In our case, we have the exponential function (2/3)^x. Notice that the base is a fraction between 0 and 1. This tells us that the function will exhibit exponential decay, meaning the value of the function decreases as 'x' increases. Let's break down the components of our specific function, f(x) = (2/3)^x. Here, the base, which is 2/3, plays a significant role in determining the function's behavior. Since the base is a fraction less than 1, the function is a decreasing exponential function. This means as the value of x increases, the value of f(x) will decrease. Understanding this fundamental characteristic is crucial for accurately completing the table of values. Moreover, the exponent 'x' dictates how many times we multiply the base by itself. For instance, if x = 2, we would calculate (2/3) * (2/3). This basic understanding of how the function works will make it much easier to compute the values in the table.

Knowing that the function represents exponential decay gives us a head start in predicting the outcomes. As we substitute different values for 'x', we'll see how the function value changes. This anticipatory understanding helps in identifying potential errors and verifying the correctness of our calculations. So, as we move forward, keep in mind that the exponential function f(x) = (2/3)^x is a decreasing function, and this knowledge will guide our computations.

Completing the Table of Values

Now, let's get to the fun part: completing the table! We have the following table to fill:

We need to find the values of d, e, and f by substituting the given 'x' values into our function (2/3)^x.

Calculating 'd' for x = -1

First, let's find 'd' when x = -1. We substitute -1 into our function: (2/3)^-1. Remember that a negative exponent means we take the reciprocal of the base raised to the positive exponent. So, (2/3)^-1 is the same as (3/2)^1, which equals 3/2. Therefore, d = 3/2. Understanding negative exponents is crucial here. When you see a negative exponent, think reciprocal. This simple trick will help you avoid many common mistakes. It's also worth noting that the reciprocal of 2/3 is indeed 3/2, which aligns with our calculation. This step-by-step approach ensures accuracy and a clear understanding of the process. Calculating 'd' correctly sets the stage for the subsequent calculations and reinforces the importance of exponent rules.

Moreover, recognizing that 3/2 is greater than 1 is consistent with our understanding of exponential decay. When x = -1, we are essentially moving in the opposite direction of the decay, resulting in a value greater than the function's value at x = 0. This kind of logical verification is essential for building confidence in our solutions.

Calculating 'e' for x = 0

Next, let's calculate 'e' when x = 0. We substitute 0 into the function: (2/3)^0. Anything raised to the power of 0 is 1. So, e = 1. This is a fundamental rule in exponents, and it's important to remember. Any non-zero number raised to the power of zero always equals one. This rule simplifies many calculations, and in our case, it directly gives us the value of 'e'. Calculating 'e' is straightforward, but it reinforces a key concept in exponential functions. Understanding this rule is essential for dealing with any exponential expression or function. It's a building block for more complex calculations and is widely used in various mathematical contexts.

Also, keep in mind that the point (0, 1) is a common characteristic of many exponential functions. It represents the initial value or the starting point of the exponential growth or decay. In our specific function, this further confirms that we are on the right track. Recognizing these patterns and characteristics is a valuable skill in mathematics.

Calculating 'f' for x = 2

Now, let's find 'f' when x = 2. We substitute 2 into the function: (2/3)^2. This means we multiply 2/3 by itself: (2/3) * (2/3) = 4/9. So, f = 4/9. When you have a fractional base raised to a positive exponent, you simply raise both the numerator and the denominator to that exponent. Calculating 'f' involves straightforward multiplication but is a crucial step in understanding how the function behaves for positive values of 'x'. Make sure to square both the 2 and the 3 to get the correct result. This basic calculation helps us see how the function's value decreases as x increases, which is consistent with the decay nature of our exponential function.

Also, note that 4/9 is less than 1, which is in line with our earlier observation about exponential decay. As x increases from 0 to 2, the function's value decreases from 1 to 4/9. This pattern reinforces our understanding of the function's behavior and helps us verify the accuracy of our calculations. By observing these trends, we can gain a deeper insight into the properties of exponential functions.

The Completed Table

Okay, guys, we've done all the calculations! Let's fill in the completed table:

We have successfully completed the table of values for the exponential function (2/3)^x. This table provides a clear picture of how the function behaves for different values of 'x'. We can see the exponential decay in action as the values decrease as 'x' increases. The table also serves as a valuable reference for graphing the function or further analysis. The completed table is a visual representation of the function's behavior and is a crucial tool for understanding exponential functions.

By having this table, we can now plot these points on a graph and visualize the curve of the exponential function. This graphical representation provides additional insights into the function's characteristics, such as its y-intercept, asymptotes, and rate of decay. Therefore, the table is not just a set of values but a gateway to a deeper understanding of the function.

Conclusion

And there you have it! Completing the table of values for the exponential function (2/3)^x is a breeze once you understand the basics of exponents and function evaluation. Remember, practice makes perfect, so keep working on these types of problems to master the concept. We've covered the importance of understanding exponential functions, applying exponent rules, and systematically filling in the table. By following these steps, you can confidently tackle any similar problem.

Remember, guys, exponential functions are all around us, so understanding them is super important. Keep practicing, and you'll become a pro in no time! Whether you're dealing with compound interest, population growth, or any other real-world scenario involving exponential change, the skills you've learned here will be invaluable. So, keep exploring, keep learning, and most importantly, keep having fun with math!