Decoding The Exponential Decay: Graphing F(x) = 60(1/3)^x

Hey guys! Let's dive into the fascinating world of exponential functions and figure out the best way to describe the graph of the function $f(x)=60 left(rac{1}{3} ight)^x$. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step and make sure you understand what's going on. This function, $f(x)=60 left(rac{1}{3} ight)^x$, represents exponential decay. This means the value of the function decreases as x increases. Think of it like a bouncing ball that loses height with each bounce, or a radioactive substance that gradually loses its mass.

Understanding Exponential Functions and Their Graphs

So, what exactly is an exponential function? In simple terms, it's a function where the variable (in this case, x) is in the exponent. This creates a curve that either increases very rapidly (exponential growth) or decreases very rapidly (exponential decay). The general form of an exponential function is $f(x) = a * b^x$, where:

• a is the initial value (the value of the function when x = 0).
• b is the base. If 0 < b < 1, you have exponential decay. If b > 1, you have exponential growth.
• x is the exponent, the variable that changes.

Let's get back to our function, $f(x) = 60 left(rac{1}{3} ight)^x$. Here, a = 60 and b = 1/3. Since b (1/3) is between 0 and 1, we know this is an exponential decay function. This means the graph will start high and curve downwards, getting closer and closer to the x-axis but never actually touching it. This line that the graph approaches is called an asymptote.

Now, let's explore the given options to see which one best describes our function's graph. We'll analyze each one critically. We need to be careful and go through each option methodically. Keep your eyes on the prize, and let's choose the best one. Remember, we are looking for the most accurate and complete description of the function's graphical behavior.

Option A: Analyzing the Incorrect Description

Let's break down option A: "The graph has an initial value of 20, and each successive term is determined by subtracting 1/3." This statement is completely off base for an exponential function. While it mentions an initial value (which is good), the second part about subtracting 1/3 is incorrect. Exponential functions don't work by adding or subtracting a constant value. Instead, they change by a factor (in our case, the base, 1/3). Moreover, the initial value of the given function is not 20 but 60, making the first part of this option also inaccurate. So, we can immediately cross out this option because it's fundamentally flawed.

Exponential functions behave differently than what's described in option A. They either grow or decay by a constant factor, not by adding or subtracting a constant amount. The graph does not follow a linear pattern where we subtract 1/3 each time; instead, the y-values decrease by a factor of 1/3 for every increment of x. The rate of change isn't constant; it is relative to the current value. This concept is the defining feature of exponential functions. This option describes a linear function rather than an exponential function.

Option B: The Correct Description of the Graph

Let's move on to Option B, which we'll analyze and confirm to ensure it is the right answer. Option B reads: "The graph has an initial value of 60, and each successive term is determined by multiplying by 1/3." This sounds promising because it mentions multiplication, a key characteristic of exponential functions. Let's delve deeper into this statement to ensure its correctness.

This option directly reflects the behavior of the given exponential function. The initial value is indeed 60, which is the value of the function when x = 0. Furthermore, each successive term is generated by multiplying the previous term by 1/3. This is because the base of our exponential function is 1/3. This is the heart of exponential decay: the function's value decreases by a factor (in this case, 1/3) with each increment of x. So, the graph starts at 60 (when x = 0), and as x increases, each subsequent y-value is obtained by multiplying the previous value by 1/3. This creates a curve that decreases rapidly initially and then gradually approaches the x-axis, the function's asymptotic behavior. So, this option perfectly describes the characteristics of the given exponential function, confirming that it is the correct answer. The function shows decay because the base is less than 1. The multiplying by 1/3 means the value decreases, getting closer and closer to zero.

Comparing Options: The Best Choice

By carefully examining both options, we can confidently conclude that Option B is the correct answer. Option A presents an incorrect description of the function. It confuses the behavior of exponential functions with linear functions. The key to answering this question is understanding how exponential functions work. We needed to identify the initial value and the factor by which the function changes. Option B accurately captures these elements. The statement is a correct description of the graph because it identifies both the initial value and the constant factor.

Summary: Exponential Decay in Action

In summary, the graph of $f(x)=60 left(rac{1}{3} ight)^x$ represents an exponential decay function. Its initial value is 60, and each subsequent term is obtained by multiplying the previous term by 1/3. The graph starts high (at 60) and decreases rapidly at first, then gradually tapers off, getting closer and closer to the x-axis. Keep in mind that understanding these fundamental concepts is crucial for grasping how exponential functions behave. Always pay attention to the initial value and the base of the exponential function, as they define the growth or decay pattern.

Key Takeaways

• Exponential functions either increase (growth) or decrease (decay).
• The base of the exponential function determines whether it grows or decays.
• Exponential decay means the function value decreases over time.
• The graph gets closer to the x-axis without ever touching it (asymptotic behavior).

Alright, guys, that's it for today! I hope you found this breakdown helpful. Keep practicing, and you'll master exponential functions in no time. If you have any more questions, feel free to ask!