Exponential Functions: Decoding The Tables

Welcome, math enthusiasts! Today, we're diving deep into the fascinating world of exponential functions, specifically those that follow the form $y = b^x$ where the base $b$ is between 0 and 1 (i.e., $0 < b < 1$). Understanding these functions is crucial because they model a wide array of real-world phenomena, from radioactive decay to the cooling of objects. When we talk about an exponential function with a base between 0 and 1, we're referring to a function where the value of $y$ decreases as $x$ increases. This is in contrast to exponential functions where the base is greater than 1, which show growth. Let's break down what makes a table represent such a function and how to spot it.

Identifying Exponential Functions with $0 < b < 1$

To determine if a table represents an exponential function of the form $y = b^x$ with $0 < b < 1$, we need to examine the relationship between the $x$ and $y$ values. First and foremost, let's recall the fundamental definition of an exponential function. In the form $y = b^x$, the variable $x$ is in the exponent. The base, $b$, is a positive constant that is not equal to 1. For our specific case, we are interested in bases where $0 < b < 1$. This condition means that as $x$ gets larger, $y$ gets smaller. Think about simple fractions like $1/2$. If you raise $1/2$ to increasingly larger powers (e.g., $(1/2)^1, (1/2)^2, (1/2)^3$), the result gets closer and closer to zero. Conversely, as $x$ becomes more negative, $y$ will increase. For instance, $(1/2)^{-2}$ is $2^2$, which equals 4. So, a key characteristic to look for in your table is a decreasing trend in the $y$ values as the $x$ values increase. This decreasing trend is the visual hallmark of an exponential function with a base between 0 and 1.

Furthermore, for a table to represent any function, each $x$-value must correspond to exactly one $y$-value. This is a fundamental rule of functions. When we're looking at exponential functions, specifically $y=b^x$, we expect a consistent multiplicative relationship between consecutive $y$-values when the $x$-values change by a constant amount. If the $x$-values in your table increase by a constant step (e.g., incrementing by 1), then the ratio of consecutive $y$-values should be constant. This constant ratio is precisely the base $b$. For example, if you have points $(x_1, y_1)$ and $(x_2, y_2)$, and $x_2 = x_1 + 1$, then $y_2 / y_1$ should equal $b$. If $x_2 = x_1 + k$ for some constant $k$, then $y_2 / y_1$ should equal $b^k$. Since we are looking for $0 < b < 1$, this ratio will be a fraction between 0 and 1. Let's consider the provided tables to see which one fits these criteria. We need to carefully check each pair of $x$ and $y$ to ensure it aligns with the properties of an exponential function where $0 < b < 1$. This involves not only observing the decreasing trend but also verifying the consistent multiplicative relationship.

Analyzing the Provided Tables

Let's scrutinize the tables presented to identify the one that accurately represents an exponential function of the form $y = b^x$ with $0 < b < 1$. We'll examine each table meticulously, looking for the characteristic decreasing trend and the consistent ratio between successive $y$-values for constant increments in $x$. It's important to be thorough, as sometimes tables can be misleading or contain errors. We're hunting for that specific pattern where $y$ gets smaller as $x$ gets larger, and where this decrease follows a strict mathematical rule governed by a base $b$ between 0 and 1.

Table A Analysis:

Let's start with Table A. The $x$ values are -3, -2, -2, 0, 1, 2. Wait a minute! We already see a problem here. The $x$-value -2 appears twice, with two different $y$-values: 1/9 and 1/3. This violates the fundamental definition of a function, which states that each input ($x$) must have exactly one output ($y$). Therefore, Table A cannot represent a function, let alone an exponential function. We can immediately discard this table. It's a good reminder to always check the basic definition of a function first!

Table B Analysis:

Now, let's turn our attention to Table B. The $x$ values are -3, -2, -1, 0, 1, 2. These are increasing by a constant step of 1. The corresponding $y$ values are 27, 9, 3, 1, 1/3, 1/9. We observe that as $x$ increases, $y$ is decreasing. This is promising! Now, let's check the ratios of consecutive $y$-values:

• $y_2 / y_1 = 9 / 27 = 1/3$
• $y_3 / y_2 = 3 / 9 = 1/3$
• $y_4 / y_3 = 1 / 3 = 1/3$
• $y_5 / y_4 = (1/3) / 1 = 1/3$
• $y_6 / y_5 = (1/9) / (1/3) = (1/9) * 3 = 3/9 = 1/3$

The ratio between consecutive $y$-values is consistently $1/3$. This means that the base $b$ is $1/3$. Since $0 < 1/3 < 1$, this table perfectly represents an exponential function of the form $y = b^x$ with $0 < b < 1$. Specifically, it represents the function $y = (1/3)^x$. Let's verify a few points. If $x=0$, $y = (1/3)^0 = 1$. This matches the table. If $x=2$, $y = (1/3)^2 = 1/9$. This also matches. If $x=-3$, $y = (1/3)^{-3} = 3^3 = 27$. This matches as well! Thus, Table B is the correct representation.

Table C Analysis:

Let's look at Table C. The $x$ values are -3, -2, -1, 0, 1, 2. Again, these are increasing by a constant step of 1. The corresponding $y$ values are 1/27, 1/9, 1/3, 1, 3, 9. As $x$ increases, $y$ is increasing. This is the opposite of what we expect for an exponential function with $0 < b < 1$. In fact, this table represents an exponential function with a base $b > 1$. Let's check the ratios to confirm:

• $y_2 / y_1 = (1/9) / (1/27) = (1/9) * 27 = 3$
• $y_3 / y_2 = (1/3) / (1/9) = (1/3) * 9 = 3$
• $y_4 / y_3 = 1 / (1/3) = 3$
• $y_5 / y_4 = 3 / 1 = 3$
• $y_6 / y_5 = 9 / 3 = 3$

The constant ratio is 3. This means Table C represents the function $y = 3^x$, which is an exponential function, but its base $b=3$ is not between 0 and 1. Therefore, Table C does not fit our criteria.

Conclusion: Spotting the Decreasing Trend

In summary, when you're tasked with identifying an exponential function of the form $y = b^x$ where $0 < b < 1$, the most immediate visual cue is a decreasing trend in $y$-values as $x$-values increase. This is because raising a number between 0 and 1 to successively larger powers results in progressively smaller numbers. Additionally, you must confirm that the table adheres to the definition of a function (each $x$ maps to only one $y$) and that there's a consistent multiplicative relationship between $y$-values for constant steps in $x$. The constant ratio you find will be your base $b$, and you must verify that $0 < b < 1$. Our analysis clearly showed that Table A fails the function test, Table C represents an exponential function but with a base greater than 1, and Table B accurately depicts an exponential function with a base between 0 and 1. It’s all about looking for that characteristic decay! Remember, these functions are fundamental to understanding many natural processes, making their identification a valuable skill in mathematics and science.

For further exploration into the properties and applications of exponential functions, you can visit the Khan Academy, a fantastic resource for learning mathematics.