Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of exponential functions, specifically focusing on how to graph the function h(x) = 8 * (3/4)^x. This might seem daunting at first, but trust me, with a step-by-step approach and a little understanding, you'll be graphing exponential functions like a pro in no time! So, grab your graph paper (or your favorite graphing software), and let's get started!
Understanding Exponential Functions
Before we jump into the specifics of h(x) = 8 * (3/4)^x, let's quickly recap what exponential functions are all about. At their core, exponential functions are functions where the independent variable (x in our case) appears as an exponent. The general form of an exponential function is f(x) = a * b^x, where a is the initial value (the y-intercept) and b is the base (the growth or decay factor). Understanding the components of exponential functions is crucial for accurately graphing them. The base, b, plays a significant role in determining the function's behavior. If b is greater than 1, the function represents exponential growth; if b is between 0 and 1, it represents exponential decay. The initial value, a, simply scales the function vertically. Think of it as stretching or compressing the graph along the y-axis. Knowing these basic principles, we can start dissecting our function, h(x) = 8 * (3/4)^x, and see how it fits into this framework. In our specific case, a is 8 and b is 3/4. Since 3/4 is between 0 and 1, we know we're dealing with an exponential decay function. This means that as x increases, the value of h(x) will decrease, approaching zero but never actually reaching it. This horizontal line that the function approaches is called the asymptote, and it's a key feature to consider when graphing exponential functions. Identifying the asymptote helps you understand the function's long-term behavior. The initial value of 8 tells us that the graph will intersect the y-axis at the point (0, 8). This provides us with a starting point for plotting the graph. Remember, guys, exponential functions are powerful tools for modeling real-world phenomena like population growth, radioactive decay, and compound interest. So, understanding how to graph them is not just an academic exercise; it's a valuable skill that can be applied in various fields.
Analyzing h(x) = 8 * (3/4)^x
Now, let's break down the function h(x) = 8 * (3/4)^x piece by piece. As we discussed earlier, this is an exponential function in the form f(x) = a * b^x. Identifying the 'a' and 'b' values is the first step to understanding the function's behavior. In this case, a = 8 and b = 3/4. The value of a, which is 8, represents the initial value or the y-intercept of the graph. This means the graph will pass through the point (0, 8). The y-intercept is a crucial reference point for sketching the graph. The value of b, which is 3/4, is the base of the exponent. Since 3/4 is between 0 and 1, this indicates that the function represents exponential decay. Exponential decay means the function's value decreases as x increases. This means that as we move from left to right along the x-axis, the graph will descend. The function will approach the x-axis (y = 0) but never actually touch it. This horizontal line, y = 0, is the asymptote of the function. Understanding the asymptote is essential for drawing the graph accurately. The asymptote acts as a guide, showing the line the function approaches but never crosses. To further analyze the function, we can consider its domain and range. The domain of this function is all real numbers, meaning we can plug in any value for x. The domain of an exponential function is typically all real numbers. However, the range is limited. Since the function approaches 0 but never reaches it, and it starts at 8 and decreases, the range is all positive real numbers less than or equal to 8, or (0, 8]. The range is determined by the function's behavior and asymptote. Understanding these key characteristics – the initial value, the base, the asymptote, the domain, and the range – provides a solid foundation for graphing the function. A thorough analysis helps you anticipate the graph's shape and behavior. We can now move on to plotting specific points and connecting them to create the graph.
Creating a Table of Values
To accurately graph the function h(x) = 8 * (3/4)^x, a crucial step is creating a table of values. Creating a table of values helps you plot accurate points on the graph. This involves choosing a range of x values and calculating the corresponding h(x) values. Selecting a mix of positive, negative, and zero values for x provides a comprehensive view of the function's behavior. Choosing diverse x-values reveals different aspects of the function's curve. Let's start with some simple values like x = -2, -1, 0, 1, and 2. We'll calculate h(x) for each of these values:
- x = -2: h(-2) = 8 * (3/4)^(-2) = 8 * (4/3)^2 = 8 * (16/9) = 128/9 ≈ 14.22
- x = -1: h(-1) = 8 * (3/4)^(-1) = 8 * (4/3) = 32/3 ≈ 10.67
- x = 0: h(0) = 8 * (3/4)^(0) = 8 * 1 = 8
- x = 1: h(1) = 8 * (3/4)^(1) = 8 * (3/4) = 6
- x = 2: h(2) = 8 * (3/4)^(2) = 8 * (9/16) = 9/2 = 4.5
Now, let's add a few more values to get a better sense of the curve, say x = 3 and x = -3:
- x = 3: h(3) = 8 * (3/4)^(3) = 8 * (27/64) = 27/8 = 3.375
- x = -3: h(-3) = 8 * (3/4)^(-3) = 8 * (4/3)^3 = 8 * (64/27) = 512/27 ≈ 18.96
We now have a table of values:
| x | h(x) |
|---|---|
| -3 | ≈ 18.96 |
| -2 | ≈ 14.22 |
| -1 | ≈ 10.67 |
| 0 | 8 |
| 1 | 6 |
| 2 | 4.5 |
| 3 | 3.375 |
This table provides coordinates to plot on the graph. These points will help us sketch the graph of the function. More points create a more accurate representation of the curve. Remember, the more points we plot, the more accurate our graph will be. This table gives us a clear picture of how the function decays as x increases. The table highlights the decaying nature of the exponential function. With these values, we're ready to plot the points on a graph and connect them to visualize the function.
Plotting the Points and Sketching the Graph
With our table of values in hand, we're ready to plot the points on a coordinate plane and sketch the graph of h(x) = 8 * (3/4)^x. Plotting points is the visual bridge between the table and the graph. First, draw your x and y axes. Clear and accurately drawn axes are essential for a good graph. Remember to label them! Now, carefully plot each point from our table: (-3, ≈18.96), (-2, ≈14.22), (-1, ≈10.67), (0, 8), (1, 6), (2, 4.5), and (3, 3.375). Accurate plotting ensures the graph represents the function correctly. You'll notice that the points are decreasing as you move from left to right, which confirms our earlier analysis that this is an exponential decay function. The plotted points should visually confirm the function's expected behavior. Now, the fun part: sketching the graph! Start by drawing a smooth curve that connects the plotted points. A smooth curve captures the essence of the exponential function. Remember that the graph will approach the x-axis (y = 0) as x increases, but it will never actually touch or cross it. This is the asymptote we discussed earlier. The asymptote guides the graph's behavior as x approaches infinity. Similarly, as x decreases (becomes more negative), the graph will increase rapidly. The rapid increase on one side and the asymptote on the other are hallmarks of exponential functions. Your graph should show a smooth, decreasing curve that starts high on the left side, passes through the point (0, 8), and gradually approaches the x-axis as it moves to the right. The final graph should visually represent the function's key characteristics. If your graph looks like this, congratulations! You've successfully graphed the exponential function h(x) = 8 * (3/4)^x. A well-drawn graph provides a visual understanding of the function. Remember, practice makes perfect. The more you graph exponential functions, the more comfortable you'll become with recognizing their characteristics and sketching their graphs.
Key Features of the Graph
After plotting the points and sketching the graph of h(x) = 8 * (3/4)^x, it's crucial to identify and understand the key features of the graph. Identifying key features solidifies your understanding of the function. These features provide a comprehensive understanding of the function's behavior and characteristics. Let's start with the y-intercept. The y-intercept is the point where the graph intersects the y-axis. The y-intercept is a starting point for understanding the function's initial value. In our function, the y-intercept is (0, 8), which we knew from the initial value a = 8. This means that when x is 0, the value of h(x) is 8. Next, let's consider the asymptote. As we discussed earlier, the asymptote is a horizontal line that the graph approaches but never touches. The asymptote reveals the function's long-term behavior. For h(x) = 8 * (3/4)^x, the asymptote is the x-axis, which is the line y = 0. This means that as x gets larger and larger, the value of h(x) gets closer and closer to 0, but it never actually reaches 0. The domain of the function is the set of all possible x values that can be input into the function. The domain defines the range of possible x-values. For exponential functions like this one, the domain is all real numbers, which can be written as (-∞, ∞). This means we can plug in any value for x and get a valid output. The range of the function is the set of all possible h(x) values that the function can output. The range defines the span of possible y-values. Since the function approaches 0 but never reaches it, and it starts at 8 and decreases, the range is all positive real numbers less than or equal to 8, or (0, 8]. Understanding the range provides insights into the function's output limits. Another important feature is the decay factor. In this case, the base b = 3/4 is the decay factor. The decay factor quantifies the rate of decrease in the function's value. Since 3/4 is between 0 and 1, it confirms that this is an exponential decay function. This means that the function's value decreases as x increases. By understanding these key features – the y-intercept, the asymptote, the domain, the range, and the decay factor – you can gain a deep understanding of the behavior of the exponential function h(x) = 8 * (3/4)^x and similar functions. Analyzing these features provides a comprehensive understanding of the function's behavior.
Conclusion
Alright, guys! We've made it through the journey of graphing the exponential function h(x) = 8 * (3/4)^x. From understanding the basic principles of exponential functions to analyzing the specific components of our function, creating a table of values, plotting points, and sketching the graph, we've covered a lot of ground. Mastering the graphing process provides a powerful tool for understanding exponential functions. We also delved into the key features of the graph, including the y-intercept, asymptote, domain, range, and decay factor. Understanding key features solidifies your grasp on the function's behavior. Remember, the most important takeaway is that graphing exponential functions doesn't have to be intimidating. By breaking down the process into manageable steps and understanding the underlying concepts, you can confidently tackle any exponential function that comes your way. A step-by-step approach makes graphing exponential functions manageable. Practice is key! The more you work with exponential functions, the more intuitive they will become. So, keep graphing, keep exploring, and keep expanding your mathematical horizons. Consistent practice builds confidence and proficiency in graphing. Exponential functions are powerful tools in mathematics and have wide-ranging applications in various fields, from finance to science. Exponential functions are valuable tools with real-world applications. So, the time you invest in understanding them is well worth it. Keep up the great work, and I'll see you in the next math adventure! Continuous learning and practice are essential for mastering mathematical concepts.