Graphing And Analyzing F(x) = (3/2)(2)^x

Let's dive into the world of exponential functions, guys! In this article, we're going to break down the function f(x) = (3/2)(2)^x. We'll not only graph it but also explore its key characteristics, making sure you understand every nook and cranny of this mathematical concept. So, buckle up and let’s get started!

Understanding Exponential Functions

Before we jump straight into graphing our function, it's super important to grasp the basics of exponential functions. Think of them as the rock stars of the function world, known for their rapid growth! The general form of an exponential function is f(x) = a(b)^x, where 'a' is the initial value and 'b' is the base. The base, 'b', is a positive number not equal to 1, and it dictates whether the function will grow or decay. If b > 1, we're talking about exponential growth; if 0 < b < 1, it's exponential decay. In our specific case, f(x) = (3/2)(2)^x, we can immediately see that a = 3/2 and b = 2. Since b = 2 is greater than 1, we know we're dealing with exponential growth. This means as x increases, f(x) will increase dramatically. Understanding this fundamental behavior is the first step in truly mastering exponential functions. Now, why is this important? Well, recognizing the growth pattern helps us predict the function's behavior without even plotting points. We know it's going to shoot upwards as we move to the right on the graph. This intuition is invaluable when analyzing and comparing different functions. Plus, in the real world, exponential growth models a ton of phenomena, from population increase to compound interest, making it a seriously practical concept to understand.

Identifying Key Features of f(x) = (3/2)(2)^x

Okay, let’s zero in on our function: f(x) = (3/2)(2)^x. To really understand it, we need to pinpoint its key features. These features are like the function's fingerprints – they tell us a lot about its personality. First up, the initial value. This is the value of the function when x = 0. It's where our graph intersects the y-axis. For our function, when we plug in x = 0, we get f(0) = (3/2)(2)^0 = (3/2)(1) = 3/2. So, our initial value is 3/2 or 1.5. This tells us our graph starts at the point (0, 1.5). Next, let's talk about the base. As we mentioned earlier, the base in our function is 2. This is a crucial piece of information because it determines the rate of growth. Since the base is 2, the function will double for every increase of 1 in x. This is a pretty significant growth rate! Now, what about asymptotes? Exponential functions have a horizontal asymptote, which is a line that the graph approaches but never quite touches. In our case, the horizontal asymptote is the x-axis (y = 0). The function will get closer and closer to the x-axis as x becomes very negative, but it will never actually cross it. Another thing to consider is whether the function is increasing or decreasing. Given that our base is greater than 1, the function is strictly increasing. This means as x gets larger, f(x) also gets larger. Understanding these key features – the initial value, the base, the horizontal asymptote, and whether the function is increasing or decreasing – gives us a solid foundation for graphing and analyzing f(x) = (3/2)(2)^x.

Step-by-Step Guide to Graphing f(x) = (3/2)(2)^x

Alright, let’s get to the fun part: graphing f(x) = (3/2)(2)^x! Don't worry, it's not as intimidating as it might seem. We'll break it down into simple steps so you can nail it every time.

Step 1: Create a Table of Values. This is our trusty sidekick for plotting points. We'll choose a few values for x, both positive and negative, and calculate the corresponding f(x) values. This gives us the coordinates we need to plot on our graph. Here’s a little table to get us started:

Step 2: Plot the Points. Now, grab your graph paper (or your favorite graphing software) and plot the points from our table. Each (x, f(x)) pair becomes a point on the graph. For example, we'll plot (-2, 3/8), (-1, 3/4), (0, 3/2), (1, 3), and (2, 6).

Step 3: Draw the Curve. This is where the magic happens. Connect the points with a smooth curve. Remember, since this is an exponential function, the curve will get steeper as x increases. Also, keep in mind that the graph approaches the x-axis (y = 0) as x goes towards negative infinity, but it never actually touches it. That’s our horizontal asymptote in action!

Step 4: Add the Asymptote. To complete the graph, draw a dashed line along the x-axis (y = 0). This visually represents the horizontal asymptote, reminding us that the function gets infinitely close to this line without ever crossing it. And there you have it! You’ve successfully graphed f(x) = (3/2)(2)^x. This step-by-step approach makes graphing exponential functions a breeze. With a little practice, you’ll be graphing these functions like a pro!

Analyzing the Graph of f(x) = (3/2)(2)^x

So, we've got our graph of f(x) = (3/2)(2)^x looking all snazzy. But the real fun starts when we begin to analyze it! Looking at the graph, we can glean a ton of information about the function's behavior. First off, let’s talk about the domain and range. The domain is all the possible x-values that the function can accept, and the range is all the possible f(x) or y-values that the function can output. For our function, the domain is all real numbers. You can plug in any value for x, positive, negative, or zero, and the function will give you a valid output. However, the range is a bit more restricted. Since our function has a horizontal asymptote at y = 0 and it's an increasing function, the range is all positive real numbers (f(x) > 0). It's crucial to understand this because it tells us the function will never produce a negative value or zero. Next up, let's consider the intercepts. We already found the y-intercept when we determined the initial value. It's the point where the graph crosses the y-axis, which is (0, 3/2). As for x-intercepts, our function doesn't have any! This is because the graph approaches the x-axis but never touches it, thanks to the horizontal asymptote. Another key aspect to analyze is the function's growth. As we move from left to right on the graph, the function increases rapidly. This is characteristic of exponential growth functions. The larger the base (in our case, 2), the steeper the growth. Finally, thinking about the end behavior is important. As x approaches positive infinity, f(x) also approaches positive infinity – it shoots upwards without bound. As x approaches negative infinity, f(x) approaches 0, getting closer and closer to the x-axis. By analyzing the domain, range, intercepts, growth, and end behavior, we gain a comprehensive understanding of how the function behaves and what we can expect from it.

Real-World Applications of Exponential Functions

Okay, so we've dissected f(x) = (3/2)(2)^x inside and out. But you might be thinking,