Identifying The Graph Of Exponential Function Y=5*(1/3)^x
Key Characteristics of Exponential Decay Functions
Exponential decay functions are characterized by a base between 0 and 1. In our function, the base is 1/3, which falls squarely within this range. This base signifies that as x increases, the value of (1/3)^x decreases, causing the overall function value to diminish. The coefficient, which is 5 in our case, acts as a vertical stretch or compression factor and also represents the y-intercept of the graph. Understanding these characteristics is crucial for visually identifying the correct graph.
Analyzing the Base (1/3)
The base of the exponent dictates the rate at which the function decays. A smaller base results in a more rapid decay. Since 1/3 is a relatively small fraction, we expect the graph to show a fairly steep decline as x increases. This steepness is a visual cue that helps distinguish this graph from other exponential functions with different bases. Furthermore, the fact that the base is a fraction between 0 and 1 confirms that the function is indeed decaying, meaning the y-values will get closer and closer to zero as x increases, but they will never actually reach zero. This is a crucial aspect of exponential decay functions, and it's reflected in the graph's shape, which approaches the x-axis asymptotically.
Interpreting the Coefficient (5)
The coefficient 5 plays two significant roles. First, it vertically stretches the graph. If we were to compare this function to y=(1/3)^x, the graph of y=5(1/3)^x* would appear stretched vertically by a factor of 5. Second, and perhaps more importantly, the coefficient represents the y-intercept of the graph. When x=0, y=5(1/3)^0=51=5**. This means the graph will intersect the y-axis at the point (0, 5). This y-intercept is a key feature to look for when identifying the correct graph.
Connecting the Characteristics to the Graph
Considering both the base and the coefficient, we can paint a clear picture of the graph we're looking for. It should be a curve that starts at y=5 on the y-axis and then descends rapidly, approaching the x-axis but never touching it. The steepness of the decay should be noticeable, reflecting the small base of 1/3. This visual representation serves as a roadmap as we examine different graph options, allowing us to match the function's mathematical properties to its graphical representation. In essence, we are looking for a graph that embodies the principles of exponential decay, with a specific starting point and rate of decline dictated by the function's parameters.
Process of Graph Identification
Identifying the correct graph of an exponential function like y=5(1/3)^x* involves a systematic approach. This method focuses on key features such as the y-intercept, the direction of the curve (increasing or decreasing), and the presence of any asymptotes. Each of these elements provides a clue, and by combining these clues, we can confidently select the graph that accurately represents the function.
Step 1: Determining the Y-intercept
The y-intercept is the point where the graph intersects the y-axis. It's found by setting x=0 in the function. For y=5(1/3)^x*, when x=0, y=5(1/3)^0=51=5**. Therefore, the y-intercept is (0, 5). This means the correct graph must pass through the point (0, 5). This single point is a powerful discriminator, immediately ruling out any graphs that do not intersect the y-axis at this specific location. Identifying the y-intercept is often the first and most straightforward step in the graph identification process.
Step 2: Analyzing the Direction of the Curve
As we discussed earlier, the base of the exponent determines whether the function is increasing or decreasing. Since the base (1/3) is between 0 and 1, this is an exponential decay function. This means the graph will decrease as x increases. In visual terms, the curve will descend from left to right. This is a critical piece of information, as it narrows down the possibilities to only those graphs that exhibit a downward trend. It helps us distinguish exponential decay functions from exponential growth functions, which increase as x increases.
Step 3: Identifying Asymptotes
Exponential functions have a horizontal asymptote, which is a line that the graph approaches but never touches. For exponential decay functions, the horizontal asymptote is typically the x-axis (y=0). This is because as x approaches infinity, (1/3)^x gets closer and closer to zero, and thus y=5(1/3)^x* approaches zero as well. The graph will get infinitesimally close to the x-axis but never actually cross it. The presence of this asymptote is a hallmark of exponential functions, and it's an important feature to look for in the graph.
Step 4: Eliminating Incorrect Graphs
With the y-intercept, the direction of the curve, and the asymptote in mind, we can systematically eliminate incorrect graphs. Any graph that does not pass through (0, 5), does not decrease from left to right, or does not approach the x-axis as x increases can be discarded. This process of elimination is a powerful tool, allowing us to focus on the remaining options and make a more informed decision. By carefully comparing the characteristics of the function with the visual representations of the graphs, we can arrive at the correct answer with confidence.
Step 5: Verifying with Additional Points (If Necessary)
If after the elimination process, multiple graphs still seem plausible, it can be helpful to evaluate the function at additional points. For instance, we could calculate the value of y when x=1: y=5(1/3)^1=5/3*, which is approximately 1.67. If only one of the remaining graphs passes through or near the point (1, 1.67), that graph is likely the correct one. This step provides an extra layer of verification, ensuring that the chosen graph aligns with the function's behavior at multiple points. It reinforces our understanding of the relationship between the function's equation and its graphical representation.
Common Mistakes to Avoid
When dealing with exponential functions and their graphs, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you approach problems more effectively and avoid errors.
Misinterpreting Exponential Growth and Decay
One of the most frequent errors is confusing exponential growth and decay. Remember, if the base of the exponent is greater than 1, the function represents exponential growth, and the graph will increase from left to right. If the base is between 0 and 1, it's exponential decay, and the graph will decrease from left to right. Failing to correctly identify the nature of the function can lead to selecting a graph with the wrong trend.
Ignoring the Y-intercept
The y-intercept is a critical piece of information, but it's sometimes overlooked. As we've discussed, the y-intercept is found by setting x=0. In the function y=5(1/3)^x*, the y-intercept is (0, 5). Forgetting to calculate or incorrectly calculating the y-intercept can result in choosing a graph that doesn't even pass through the correct point on the y-axis.
Not Recognizing Asymptotes
Exponential functions have horizontal asymptotes, and understanding their presence and location is crucial for accurate graph identification. An asymptote is a line that the graph approaches but never touches. For the function y=5(1/3)^x*, the horizontal asymptote is the x-axis (y=0). Failing to recognize the asymptote can lead to selecting a graph that crosses the x-axis, which is incorrect for this type of exponential decay function.
Incorrectly Plotting Points
If you choose to verify the graph by plotting additional points, accuracy is essential. Incorrectly calculating or plotting points can lead to a misinterpretation of the graph's behavior. Double-check your calculations and ensure that your plotted points accurately reflect the function's values at specific x values. This step is particularly important if you're working without a graphing calculator or software.
Overlooking the Scale of the Axes
The scale of the axes can sometimes be misleading. Graphs can be stretched or compressed, making it difficult to accurately assess the function's behavior. Pay close attention to the units on the x and y axes. A graph that appears to have a y-intercept of 5 might actually have a different y-intercept if the scale is not consistent. Always consider the scale when interpreting graphs.
Neglecting the Steepness of the Curve
The steepness of the curve provides valuable information about the rate of growth or decay. A smaller base in an exponential decay function results in a steeper decline. If you overlook the steepness of the curve, you might choose a graph that represents a different rate of decay. In the case of y=5(1/3)^x*, the decay should be fairly rapid, reflecting the base of 1/3. Paying attention to this visual detail can help you distinguish between graphs with different decay rates.
By being mindful of these common mistakes and implementing a systematic approach to graph identification, you can improve your accuracy and confidence when working with exponential functions.
Conclusion
In summary, identifying the graph that represents the function y=5(1/3)^x* involves understanding the characteristics of exponential decay functions, systematically analyzing key features such as the y-intercept and asymptotes, and avoiding common mistakes. By recognizing that this function exhibits exponential decay due to its base being between 0 and 1, and by noting the y-intercept at (0, 5) and the horizontal asymptote at y=0, we can effectively narrow down the possibilities. The steepness of the curve also provides a crucial visual cue, reflecting the relatively rapid decay rate associated with the base of 1/3. This comprehensive approach ensures that the selected graph accurately portrays the function's mathematical properties.
Remember, the process of graph identification is not just about finding the right answer; it's about developing a deeper understanding of the relationship between functions and their graphical representations. Each step, from determining the y-intercept to analyzing the asymptote, reinforces the connection between algebraic expressions and visual patterns. This understanding is invaluable not only in mathematics but also in various fields that utilize graphical data analysis. So, approach each graph identification problem as an opportunity to hone your analytical skills and deepen your appreciation for the visual language of mathematics.
