Exponential Function Equation In Sequence Notation

Dec 3, 2025 by Alex Johnson 51 views

Let's dive into the world of exponential functions and learn how to determine their equations when given a set of ordered pairs. In this article, we'll specifically tackle the problem where we're given the ordered pairs (1, 80), (2, 240), (3, 720), and (4, 2160), which represent an exponential function p(t). Our goal is to express this function in sequence notation, which means we want to find an equation of the form pₜ = ?

Understanding Exponential Functions

Before we jump into the calculations, let's quickly recap what exponential functions are all about. At their core, exponential functions describe situations where a quantity grows or decays at a constant percentage rate over time. This is different from linear functions, where the quantity changes by a constant amount. The general form of an exponential function is:

f(x) = a * bˣ

Where:

f(x) is the value of the function at x
a is the initial value (the value when x = 0)
b is the growth/decay factor (the factor by which the function changes when x increases by 1)
x is the input variable

In our case, we're using the notation p(t), so our general form will be:

p(t) = a * bᵗ

Where:

p(t) is the value of the function at time t
a is the initial value
b is the growth/decay factor
t is the input variable (time in this case)

Decoding Sequence Notation

Sequence notation is simply a way of expressing the function's output for discrete values of the input. Instead of writing p(t), we write pₜ, where t represents the position in the sequence (1st term, 2nd term, etc.). This notation is particularly useful when dealing with sequences and patterns.

So, our goal is to find an equation for pₜ that fits the given ordered pairs.

Finding the Function Equation

Now, let's get to the fun part: determining the equation for our exponential function. We have four ordered pairs: (1, 80), (2, 240), (3, 720), and (4, 2160). These pairs tell us the value of the function p(t) at different times t.

Step 1: Find the Growth Factor (b)

The key to identifying an exponential function lies in the constant ratio between consecutive y-values (or p(t) values in our case). Let's calculate the ratio between the p(t) values:

240 / 80 = 3
720 / 240 = 3
2160 / 720 = 3

Notice that the ratio is consistently 3. This tells us that our growth factor, b, is 3. Each time t increases by 1, the function value is multiplied by 3. This constant ratio is a hallmark of exponential functions.

Step 2: Find the Initial Value (a)

Now that we know b = 3, we can use one of the ordered pairs to find the initial value, a. Let's use the first ordered pair (1, 80). We know that p(1) = 80. Plugging this into our general equation:

80 = a * 3¹

80 = 3a

Divide both sides by 3:

a = 80 / 3

So, our initial value, a, is 80/3.

Step 3: Write the Function Equation in Sequence Notation

We now have all the pieces we need to write the function equation in sequence notation. We know that a = 80/3 and b = 3. Plugging these values into our general form pₜ = a * bᵗ*, we get:

pₜ = (80/3) * 3ᵗ

This is our function equation in sequence notation! It tells us the value of the function at any position t in the sequence.

Verifying the Equation

It's always a good idea to verify our equation to make sure it's correct. Let's plug in the values of t from our ordered pairs and see if we get the corresponding pₜ values.

For t = 1: p₁ = (80/3) * 3¹ = 80 (Correct!)
For t = 2: p₂ = (80/3) * 3² = (80/3) * 9 = 240 (Correct!)
For t = 3: p₃ = (80/3) * 3³ = (80/3) * 27 = 720 (Correct!)
For t = 4: p₄ = (80/3) * 3⁴ = (80/3) * 81 = 2160 (Correct!)

Our equation works perfectly for all the given ordered pairs. This gives us confidence in our solution.

Alternative Form of the Equation

While pₜ = (80/3) * 3ᵗ is a perfectly valid equation, we can simplify it further. Notice that we can rewrite 3ᵗ as 3 * 3⁽ᵗ⁻¹⁾. Substituting this into our equation:

pₜ = (80/3) * 3 * 3⁽ᵗ⁻¹⁾

The (80/3) * 3 simplifies to 80, so we have:

pₜ = 80 * 3⁽ᵗ⁻¹⁾

This is an equivalent form of the equation, and it's often preferred because it highlights the initial value (80 when t=1) and the growth factor (3).

Conclusion

We've successfully found the equation for the exponential function represented by the given ordered pairs. We started by understanding the general form of exponential functions and sequence notation. Then, we calculated the growth factor and initial value, and finally, we expressed the function in sequence notation as pₜ = (80/3) * 3ᵗ or the simplified form pₜ = 80 * 3⁽ᵗ⁻¹⁾. Remember, the key to identifying exponential functions is the constant ratio between consecutive function values. This process can be applied to any set of ordered pairs that represent an exponential relationship.

For more information on exponential functions, you can visit Khan Academy's Exponential Functions section. This is a trusted website that offers comprehensive lessons and practice exercises.

Additional Tips for Solving Exponential Function Problems

Look for a constant ratio: The most crucial step in identifying an exponential function is to check for a constant ratio between consecutive y-values. If the ratio is constant, you're likely dealing with an exponential function.
Use the general form: Always start with the general form of the exponential function, f(x) = a * bˣ (or p(t) = a * bᵗ in our case). This provides a framework for solving the problem.
Solve for the growth factor first: The growth factor (b) is usually easier to find than the initial value (a). Calculate the ratio between consecutive y-values to determine b.
Use one ordered pair to find the initial value: Once you know the growth factor, plug it into the general equation along with the x and y values from one of the ordered pairs. This will allow you to solve for the initial value (a).
Verify your equation: After finding the equation, always check it by plugging in the x values from the other ordered pairs. If the equation produces the correct y values, you've likely found the correct function.
Consider different forms of the equation: Exponential functions can be written in different forms. For example, f(x) = a * bˣ is equivalent to f(x) = a * e^(kx), where k = ln(b). Understanding these different forms can be helpful in various situations.
Practice, practice, practice: The best way to master exponential functions is to practice solving problems. Work through examples and try different variations to build your understanding and confidence.

Real-World Applications of Exponential Functions

Exponential functions aren't just abstract mathematical concepts; they appear in many real-world scenarios. Understanding exponential functions can help you make sense of these situations and even make predictions about the future. Here are some examples:

Population Growth: The growth of a population (whether it's humans, bacteria, or animals) often follows an exponential pattern. If a population grows at a constant percentage rate, its size can be modeled using an exponential function.
Compound Interest: When you invest money in an account that earns compound interest, the amount of money grows exponentially. The more frequently the interest is compounded, the faster the growth.
Radioactive Decay: Radioactive materials decay over time, meaning they lose their mass at a constant percentage rate. This decay process can be modeled using a decreasing exponential function.
Spread of Diseases: The spread of infectious diseases can sometimes follow an exponential pattern, especially in the early stages of an outbreak. This is why it's crucial to take measures to slow down the spread of a disease.
Cooling and Heating: The rate at which an object cools down or heats up can be modeled using an exponential function. This is known as Newton's Law of Cooling.

By understanding the principles of exponential functions, you can gain valuable insights into these and many other real-world phenomena. Remember that mastering this concept will be beneficial across various disciplines, not just mathematics.