Graphing Exponential Functions F(x) = (3/4)^x A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of exponential functions, specifically how to graph them by creating a table of coordinates. We'll break down the function $f(x) = (\frac{3}{4})^x$ step by step, so you can master this skill. Graphing functions might seem daunting at first, but trust me, with a systematic approach, it's totally manageable and even fun! We will explore exponential functions through creating a table of values, plotting points, and connecting them to visualize the graph. This method provides a clear understanding of how the function behaves and is an essential technique for anyone studying mathematics. Understanding exponential functions is crucial in various fields, including finance, physics, and computer science, so let's get started!

Understanding the Exponential Function

Before we jump into graphing, let's make sure we understand what an exponential function is. In simple terms, an exponential function is a function where the variable (in our case, x) is in the exponent. The general form looks like this: $f(x) = a^x$, where a is a constant called the base.

In our specific function, $f(x) = (\frac{3}{4})^x$, the base a is $\frac{3}{4}$. This is important because the base determines the overall shape and behavior of the graph. When the base is between 0 and 1, like in our case, the function represents exponential decay. This means that as x increases, the value of $f(x)$ decreases.

Exponential functions are characterized by their rapid growth or decay. They pop up everywhere in the real world, from population growth to radioactive decay. Grasping how to graph them is a fundamental skill in mathematics and has practical applications in numerous fields. Now, let's dive deep into the mechanics of graphing $f(x) = (\frac{3}{4})^x$ by using a table of coordinates. This method will allow us to plot specific points and see how the function behaves.

Constructing the Table of Coordinates

The heart of graphing any function using this method is creating a table of coordinates. We choose a set of x-values, plug them into the function, and calculate the corresponding y-values (which are the same as $f(x)$ values). These x and y pairs then become our coordinates that we'll plot on the graph. A well-chosen set of x-values can reveal the key characteristics of the function's shape and behavior.

For our function, $f(x) = (\frac{3}{4})^x$, we'll use the given x-values: -2, -1, 0, 1, and 2. These values are strategically chosen to give us a good picture of the function's behavior on both sides of the y-axis. We will methodically evaluate the function at each x-value, which is a straightforward process that involves substituting the x-value into the function and simplifying. This hands-on approach makes the concept of functions and graphs much more tangible. So, let's roll up our sleeves and calculate the y-values for each x-value!

Calculating the y-values

Now comes the fun part – plugging in the x-values and calculating the corresponding y-values. Remember, $f(x)$ is just another way of writing y, so we're essentially finding the y-coordinate for each x-coordinate.

• For x = -2: $f(-2) = (\frac{3}{4})^{-2}$. Remember that a negative exponent means we take the reciprocal of the base and raise it to the positive exponent: $f(-2) = (\frac{4}{3})^2 = \frac{16}{9} \approx 1.78$

• For x = -1: $f(-1) = (\frac{3}{4})^{-1}$. Again, we take the reciprocal: $f(-1) = \frac{4}{3} \approx 1.33$

• For x = 0: $f(0) = (\frac{3}{4})^0$. Anything (except 0) raised to the power of 0 is 1: $f(0) = 1$

• For x = 1: $f(1) = (\frac{3}{4})^1$. Anything raised to the power of 1 is just itself: $f(1) = \frac{3}{4} = 0.75$

• For x = 2: $f(2) = (\frac{3}{4})^2$. We square the fraction: $f(2) = \frac{9}{16} = 0.5625$ (approximately 0.56)

See? It's not so scary! By carefully following the rules of exponents, we can easily find the y-values for each x-value. This process highlights the power of mathematical rules and how they enable us to make precise calculations. Now that we've conquered the calculations, let's update our table and prepare to plot these points on a graph. This transformation of numerical values into visual points is what makes graphing such a powerful tool for understanding functions.

Here's our completed table:

Plotting the Points and Sketching the Graph

With our table complete, we're ready to plot the points on a coordinate plane. Each (x, y) pair in the table represents a point on the graph. Remember, the x-value tells us how far to move horizontally (left or right from the origin), and the y-value tells us how far to move vertically (up or down from the origin).

So, we'll plot the following points:

• (-2, 1.78)
• (-1, 1.33)
• (0, 1)
• (1, 0.75)
• (2, 0.56)

Once we've plotted these points, we can start to see the shape of the graph emerging. Exponential functions have a characteristic curve, and in this case, since the base is between 0 and 1, we expect to see a decreasing curve. This anticipation, based on our understanding of the function, helps us verify that our plotted points are consistent with the expected behavior.

Now, the final step is to connect the points with a smooth curve. Remember, the graph of an exponential function is a curve, not a series of straight lines. As you draw the curve, imagine infinitely many points between the ones you've plotted, smoothly transitioning from one to the next. The curve should extend beyond the plotted points, showing the function's behavior as x gets very large (positive or negative). Pay attention to the asymptotes – the lines that the graph approaches but never quite touches. In this case, the x-axis (y = 0) is a horizontal asymptote.

As we sketch the graph, we can clearly observe the exponential decay nature of the function. The graph starts high on the left and gradually decreases, approaching the x-axis as x increases. This visualization reinforces the concept of exponential decay and helps solidify our understanding of how the function behaves.

Observations and Key Features

Looking at the graph, we can make some key observations about the function $f(x) = (\frac{3}{4})^x$:

• Exponential Decay: As x increases, the value of $f(x)$ decreases. This is because the base ($\frac{3}{4}$) is between 0 and 1.
• Y-intercept: The graph crosses the y-axis at the point (0, 1). This is because any number (except 0) raised to the power of 0 is 1.
• Horizontal Asymptote: The graph approaches the x-axis (y = 0) as x gets larger, but it never actually touches it. This means that y = 0 is a horizontal asymptote.
• Domain: The domain of the function is all real numbers. We can plug in any value for x.
• Range: The range of the function is all positive real numbers (y > 0). The function never takes on a value of 0 or a negative value.

Understanding these key features is crucial for a comprehensive understanding of exponential functions. These characteristics are not unique to this particular function but are common traits of all exponential functions with a base between 0 and 1. Recognizing these features allows us to quickly analyze and interpret the behavior of such functions. The y-intercept, asymptote, domain, and range provide a complete picture of how the function operates and its limitations.

By identifying the domain and range, we define the possible input and output values of the function, which is vital in many practical applications. The asymptote tells us about the long-term behavior of the function, indicating the value that the function approaches but never reaches. Overall, examining these key features empowers us to fully understand and utilize exponential functions in various mathematical and real-world contexts.

Conclusion

So, there you have it! We've successfully graphed the exponential function $f(x) = (\frac{3}{4})^x$ by creating a table of coordinates. Remember, the key is to choose appropriate x-values, calculate the corresponding y-values, plot the points, and connect them with a smooth curve. Understanding the properties of exponential functions, such as the base, y-intercept, and asymptotes, will help you graph them more accurately and efficiently.

Graphing functions is a fundamental skill in mathematics, and mastering it opens doors to understanding more complex concepts. It's not just about plotting points; it's about visualizing relationships and understanding the behavior of functions. So, keep practicing, and you'll become a graphing pro in no time! The ability to visualize mathematical relationships is incredibly valuable, as it allows us to connect abstract equations to concrete visual representations.

Remember, each graph tells a story, and by understanding how to read and create graphs, we can unlock a deeper understanding of the mathematical world around us. Keep exploring different functions and experimenting with different graphing techniques. The more you practice, the more intuitive it will become. Happy graphing, guys! Now, go out there and conquer those exponential functions!