Understanding Function Rate Of Change From Tables

When we talk about the rate of change of a function, we're essentially asking how much the output (usually represented by 'y') changes for every unit change in the input (usually represented by 'x'). It's a fundamental concept in mathematics that helps us understand how functions behave, whether they are increasing, decreasing, or staying constant. For linear functions, this rate of change is constant and is what we call the slope. However, for other types of functions, the rate of change can vary. Today, we're going to dive deep into how to determine this rate of change, especially when our function is presented in a table format, like the one you've provided. This method is crucial for analyzing data that might not come with a neat algebraic equation but rather as a set of observations.

Let's take a closer look at the data you've presented:

Our primary goal here is to figure out if there's a consistent pattern in how 'y' changes as 'x' increases by one unit each time. In a table like this, the 'x' values are increasing by a steady increment of 1 (-1 to 0, 0 to 1, and so on). This makes our job a bit easier because we can focus directly on how the 'y' values are transforming. To do this, we'll calculate the ratio of the change in 'y' to the change in 'x' between consecutive points. Since the change in 'x' is always 1, this ratio simplifies to just the change in 'y'.

Let's calculate the change in 'y' between each pair of consecutive points:

• From x = -1 to x = 0: Change in y = y₂ - y₁ = (1/2) - (1/10) To subtract these fractions, we need a common denominator, which is 10. (1/2) = (5/10) So, (5/10) - (1/10) = 4/10 = 2/5.

• From x = 0 to x = 1: Change in y = y₃ - y₂ = (5/2) - (1/2) These fractions already have a common denominator. (5/2) - (1/2) = 4/2 = 2.

• From x = 1 to x = 2: Change in y = y₄ - y₃ = (25/2) - (5/2) Again, a common denominator. (25/2) - (5/2) = 20/2 = 10.

• From x = 2 to x = 3: Change in y = y₅ - y₄ = (125/2) - (25/2) Common denominator. (125/2) - (25/2) = 100/2 = 50.

As we can see, the changes in 'y' are not constant (2/5, 2, 10, 50). This tells us that the function is not a linear function. If it were linear, the rate of change (the slope) would be the same between every pair of points. But this doesn't mean we can't describe the rate of change; it just means the function is more complex. In cases like this, where the rate of change isn't constant, we often look for patterns in how the rate of change itself is changing. Sometimes, functions exhibit exponential growth or decay, where the rate of change is proportional to the current value of the function. Let's explore this possibility further in the next sections.

Identifying the Type of Function

Since the rate of change isn't constant, let's investigate if there's a multiplicative relationship between consecutive 'y' values. This is a common characteristic of exponential functions. In an exponential function of the form $y = a imes b^x$, the ratio of consecutive y-values (when x increases by a constant amount) is constant and equal to the base 'b'. Let's calculate these ratios for our table:

• From x = -1 to x = 0: Ratio = y₂ / y₁ = (1/2) / (1/10) Dividing by a fraction is the same as multiplying by its reciprocal. (1/2) * (10/1) = 10/2 = 5.

• From x = 0 to x = 1: Ratio = y₃ / y₂ = (5/2) / (1/2) (5/2) * (2/1) = 10/2 = 5.

• From x = 1 to x = 2: Ratio = y₄ / y₃ = (25/2) / (5/2) (25/2) * (2/5) = 50/10 = 5.

• From x = 2 to x = 3: Ratio = y₅ / y₄ = (125/2) / (25/2) (125/2) * (2/25) = 250/50 = 5.

Wow! We found a consistent ratio of 5. This is a significant finding! It means that for every one-unit increase in 'x', the 'y' value is multiplied by 5. This pattern strongly indicates that the function represented in the table is an exponential function. The constant ratio we found (5) is the base of the exponential function. So, our function has the form $y = a imes 5^x$.

Determining the Specific Exponential Function

Now that we know the function is exponential with a base of 5, we need to find the value of 'a', which is the coefficient that scales the exponential term. We can use any point from the table to solve for 'a'. Let's use the point where x = 0 and y = 1/2.

Substituting these values into our equation $y = a imes 5^x$:

1/2 = a * 5⁰

Remember that any non-zero number raised to the power of 0 is 1 (5⁰ = 1).

1/2 = a * 1

a = 1/2

So, the specific exponential function that describes the data in the table is $y = \frac{1}{2} \times 5^x$. Let's quickly verify this with another point, say x = 1, y = 5/2:

$y = \frac{1}{2} \times 5¹ = \frac{1}{2} \times 5 = \frac{5}{2}$. This matches the table!

The Rate of Change for Exponential Functions

Now, let's return to the original question: "What is the rate of change of the function described in the table?" We've established that the function is exponential, and for exponential functions, the rate of change is not constant. Instead, the rate of change increases as 'x' increases. It's not a simple