Graphing F(x)=-3^(x-2) Domain And Range Explained

Hey everyone! Today, we're diving into the fascinating world of exponential functions. Specifically, we're going to break down how to graph the function f(x) = -3^(x-2). Don't worry if it looks intimidating at first – we'll take it step by step. We'll also explore how to determine the domain and range of this function. So, grab your pencils, and let's get started!

Understanding the Parent Function

To truly understand the function f(x) = -3^(x-2), it's beneficial, but not strictly required, to first consider its parent function, which is g(x) = 3^x. The parent function serves as the foundation upon which our transformations will be built. Now, let’s dive deep into this parent function. The exponential function g(x) = 3^x is a classic example of exponential growth. As x increases, the value of g(x) increases rapidly. This fundamental behavior stems from the fact that the base, 3 in this case, is greater than 1. When graphing g(x) = 3^x, we observe several key characteristics. The graph passes through the point (0, 1) because any number raised to the power of 0 is 1 (3^0 = 1). As x increases, the graph rises steeply, demonstrating exponential growth. Conversely, as x decreases (becomes more negative), the graph approaches the x-axis but never actually touches it. This x-axis serves as a horizontal asymptote for the function. An asymptote is a line that a curve approaches but does not intersect. For g(x) = 3^x, the horizontal asymptote is the line y = 0. The domain of g(x) = 3^x is all real numbers, meaning x can take on any value from negative infinity to positive infinity. We express this in interval notation as (-∞, ∞). The range of g(x) = 3^x, however, is restricted to positive values. Since the exponential function always yields a positive output, the range is (0, ∞). Understanding these fundamental properties of the parent function g(x) = 3^x provides a crucial framework for analyzing and graphing transformations of this function, such as our target function f(x) = -3^(x-2). By recognizing the initial behavior of the exponential growth and the significance of the horizontal asymptote, we can more easily predict and interpret the transformations that occur when we introduce modifications like reflections and shifts. So, while sketching the parent function is not mandatory, it definitely gives us a solid visual reference point. Think of it as the original blueprint before we start making renovations!

Transformations: Unveiling the Graph of f(x) = -3^(x-2)

Okay, guys, now that we have a grasp of the parent function, let's tackle the transformations that turn g(x) = 3^x into f(x) = -3^(x-2). There are two key transformations at play here: a reflection and a horizontal shift. Let's break them down one by one.

First, we have the negative sign in front of the 3, which gives us -3^x. This negative sign causes a reflection across the x-axis. Imagine taking the graph of g(x) = 3^x and flipping it over the x-axis. What was once above the x-axis is now below it, and vice versa. This reflection dramatically changes the behavior of the function. Instead of exponential growth where the function increases as x increases, we now have a situation where the function decreases as x increases. The horizontal asymptote, which was y = 0 for the parent function, remains the same after the reflection because reflecting the line y = 0 across itself doesn't change it. However, the range of the function does change due to the reflection. The range of g(x) = 3^x was (0, ∞), meaning the function's values were always positive. After the reflection, the range becomes (-∞, 0), indicating that the function's values are now always negative. Next, we have the (x - 2) in the exponent. This indicates a horizontal shift. Specifically, it shifts the graph 2 units to the right. It’s important to remember that horizontal shifts work in the opposite direction of what you might initially expect. The (x - 2) term shifts the graph to the right, not the left. To visualize this, think about what value of x would make the exponent equal to zero. In the original function g(x) = 3^x, the exponent is zero when x = 0. In the transformed function f(x) = -3^(x-2), the exponent is zero when x - 2 = 0, which means x = 2. This tells us that the point that was originally at x = 0 on the parent function has been shifted to x = 2. Combining these transformations, we can see how the graph of f(x) = -3^(x-2) is created from the parent function g(x) = 3^x. First, we reflect the graph of g(x) across the x-axis, and then we shift the reflected graph 2 units to the right. This results in a graph that decreases as x increases and has a horizontal asymptote at y = 0. By understanding these transformations, we gain a powerful tool for graphing and analyzing a wide range of exponential functions. We can predict how changes in the equation will affect the graph, allowing us to quickly sketch the function and identify its key features. Now that we've deconstructed the transformations, let's put it all together to visualize the final graph.

Sketching the Graph

Alright, let's put our knowledge into action and sketch the graph of f(x) = -3^(x-2). To do this effectively, we'll follow a step-by-step approach, leveraging our understanding of transformations. First, let's identify a few key points on the graph. These points will serve as anchors to guide our sketch and ensure accuracy. A good starting point is to find the value of the function when x = 2. This is particularly helpful because it corresponds to the horizontal shift we discussed earlier. Plugging in x = 2 into the function, we get f(2) = -3^(2-2) = -3^0 = -1. So, the point (2, -1) lies on the graph. This point is crucial because it represents the shifted version of the point (0, 1) on the parent function g(x) = 3^x after the reflection and horizontal shift. Next, let's consider what happens as x gets larger. As x increases, the exponent (x - 2) also increases. This means that 3^(x-2) becomes a larger and larger positive number. However, because of the negative sign in front, -3^(x-2) becomes a larger and larger negative number. Therefore, as x goes towards infinity, f(x) goes towards negative infinity. This indicates that the graph will continue to decrease rapidly as we move to the right. On the other hand, let's examine what happens as x gets smaller (becomes more negative). As x decreases, the exponent (x - 2) becomes a larger negative number. This means that 3^(x-2) becomes a smaller and smaller positive number, approaching zero. Consequently, -3^(x-2) also approaches zero, but from the negative side. This behavior tells us that the x-axis (the line y = 0) acts as a horizontal asymptote for the function. The graph will get closer and closer to the x-axis as x decreases, but it will never actually touch or cross it. Now that we have a good understanding of the function's behavior and some key points, we can sketch the graph. Start by plotting the point (2, -1). Then, draw a smooth curve that passes through this point and approaches the x-axis as x decreases. As x increases, the curve should decrease rapidly, moving further and further away from the x-axis in the negative direction. Remember to keep in mind the horizontal asymptote at y = 0. The graph should get very close to this line but never intersect it. The resulting sketch should resemble a reflected and shifted exponential decay curve. It starts close to the x-axis on the left side, passes through (2, -1), and then plunges downwards rapidly as we move to the right. By carefully considering the transformations and identifying key points, we can create an accurate sketch of the graph of f(x) = -3^(x-2). This visual representation provides valuable insights into the function's behavior and characteristics.

Domain of f(x) = -3^(x-2)

Now, let's determine the domain of f(x) = -3^(x-2). Remember, the domain refers to all possible input values (x-values) for which the function is defined. For exponential functions, the domain is generally all real numbers, unless there are restrictions imposed by other components of the function (like a square root or a logarithm). In this case, f(x) = -3^(x-2) doesn't have any such restrictions. We can plug in any real number for x, and the function will produce a valid output. There are no denominators that could be zero, no square roots of negative numbers, and no logarithms of non-positive numbers to worry about. Therefore, the domain of f(x) = -3^(x-2) is all real numbers. In interval notation, we express this as (-∞, ∞). This means that x can take on any value from negative infinity to positive infinity. The graph of the function extends infinitely in both the left and right directions along the x-axis, confirming that there are no gaps or breaks in the domain. Understanding the domain of a function is crucial because it tells us the range of input values for which the function is meaningful. In practical applications, the domain might be limited by real-world constraints, such as physical limitations or resource availability. However, in the abstract mathematical context, the domain of f(x) = -3^(x-2) is simply all real numbers.

Range of f(x) = -3^(x-2)

Finally, let's discuss the range of f(x) = -3^(x-2). The range represents all possible output values (y-values) that the function can produce. To determine the range, we need to consider the transformations that have been applied to the parent function g(x) = 3^x. Recall that the parent function g(x) = 3^x has a range of (0, ∞), meaning its output values are always positive and greater than zero. However, our function f(x) = -3^(x-2) has undergone two key transformations: a reflection across the x-axis and a horizontal shift. The horizontal shift does not affect the range of the function. Shifting the graph left or right simply moves the function along the x-axis, but it doesn't change the set of possible y-values. The reflection across the x-axis, however, has a significant impact on the range. The reflection flips the graph over the x-axis, which means that the positive y-values of the parent function become negative y-values in the transformed function. As a result, the range of f(x) = -3^(x-2) becomes (-∞, 0). This means that the output values of the function are always negative and less than zero. The function can take on any negative value, but it will never be zero or positive. This is because the exponential term 3^(x-2) is always positive, and the negative sign in front ensures that the entire expression is negative. Furthermore, the horizontal asymptote at y = 0 reinforces the fact that the function will never actually reach zero. The graph approaches the x-axis as x decreases, but it never touches or crosses it. Understanding the range of a function is just as important as understanding its domain. The range tells us the set of possible output values that we can expect from the function. In practical applications, the range might represent a set of physical measurements, financial values, or other quantities that are relevant to the problem being modeled. By considering the transformations and the horizontal asymptote, we have successfully determined that the range of f(x) = -3^(x-2) is (-∞, 0).

Wrapping Up

So, there you have it! We've successfully graphed the exponential function f(x) = -3^(x-2), identified its key transformations, and determined its domain and range. We learned that the domain is all real numbers, represented as (-∞, ∞), while the range is all negative numbers, represented as (-∞, 0). Remember, understanding these concepts will help you tackle more complex functions in the future. Keep practicing, and you'll become a graphing pro in no time! Hope this helped, guys!