Graphing The Sequence F(x+1) = (2/3)f(x) With Initial Value 108

What is the graph of the sequence defined by the function f(x+1) = (2/3)f(x) if the initial value of the sequence is 108?

This article delves into the intricacies of graphing a sequence defined by the recursive function f(x+1) = (2/3)f(x), given an initial value of 108. We'll explore the mathematical concepts behind this sequence, its properties, and how to effectively represent it graphically. Understanding these concepts is crucial for anyone studying sequences, functions, and their graphical representations. The given function represents a geometric sequence, which is a fundamental concept in mathematics with applications in various fields, including finance, physics, and computer science. By understanding the graph of this sequence, we can gain insights into its behavior and make predictions about its future terms. Furthermore, the process of graphing this sequence involves applying several important mathematical principles, such as identifying the common ratio, calculating terms, and plotting points on a coordinate plane. These skills are essential for mathematical problem-solving and analysis.

Decoding the Recursive Function

At its core, the function f(x+1) = (2/3)f(x) defines a geometric sequence. Let's break down what this means. A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio. In our case, the common ratio is 2/3. This means that each term in the sequence is two-thirds of the previous term. The initial value of the sequence, given as 108, is the starting point from which we generate the rest of the terms. This initial value is often denoted as f(0) or a, and it plays a crucial role in determining the behavior of the sequence. Understanding the concept of a common ratio is essential for working with geometric sequences. It allows us to predict how the terms of the sequence will change over time. In this case, because the common ratio is less than 1, we know that the terms of the sequence will decrease as we move further along the sequence. This type of sequence is called a decreasing geometric sequence. We can also express this sequence in an explicit form, which gives us a direct formula for calculating any term in the sequence without having to calculate all the previous terms. The explicit form of a geometric sequence is given by f(x) = a * r^x, where a is the initial value and r is the common ratio. In our case, the explicit form would be f(x) = 108 * (2/3)^x. This explicit form allows us to quickly calculate the value of any term in the sequence, such as the 10th term or the 100th term.

Calculating the Sequence Terms

To visualize the sequence, we need to calculate the first few terms. We start with the initial value, f(0) = 108. To find the next term, f(1), we use the recursive formula: f(1) = (2/3) * f(0) = (2/3) * 108 = 72. We can continue this process to find subsequent terms. For instance, f(2) = (2/3) * f(1) = (2/3) * 72 = 48. Similarly, f(3) = (2/3) * f(2) = (2/3) * 48 = 32, and f(4) = (2/3) * f(3) = (2/3) * 32 = 64/3 ≈ 21.33. Calculating these terms allows us to see the pattern of the sequence. We can observe that the terms are decreasing, as expected, due to the common ratio being less than 1. The more terms we calculate, the clearer the pattern becomes. We can use these calculated terms to create a table of values, which can then be used to plot the points on a graph. This table of values will show the relationship between the term number (x) and the value of the term (f(x)). By examining the table, we can further analyze the behavior of the sequence and make predictions about its long-term trend. For example, we can see that the terms are getting closer and closer to zero, which is a characteristic of a decreasing geometric sequence with a common ratio between 0 and 1. This process of calculating terms and analyzing the pattern is a fundamental skill in working with sequences and functions.

Plotting the Graph

Now, let's translate these numerical values into a graphical representation. We'll use a coordinate plane, where the x-axis represents the term number (x) and the y-axis represents the value of the term (f(x)). We plot the points we calculated earlier: (0, 108), (1, 72), (2, 48), (3, 32), and (4, 64/3). Each point represents a term in the sequence. For example, the point (0, 108) represents the initial value of the sequence, and the point (1, 72) represents the second term in the sequence. These points give us a visual representation of how the sequence changes over time. When plotting these points, it's important to choose an appropriate scale for the axes. Since the terms are decreasing, we need to make sure that the y-axis extends high enough to include the initial value of 108. The scale should also be chosen so that the points are clearly visible and the overall shape of the graph is easy to see. It's important to note that we only plot discrete points, not a continuous line, because the sequence is defined only for integer values of x (i.e., the term numbers). This is a key difference between the graph of a sequence and the graph of a continuous function. The points we plot form a distinct pattern. They fall along a curve that gradually approaches the x-axis. This curve represents the overall trend of the sequence. The fact that the points do not form a straight line indicates that the sequence is not arithmetic (where the difference between consecutive terms is constant), but rather geometric (where the ratio between consecutive terms is constant).

Analyzing the Graph and Identifying Key Characteristics

Observing the plotted points, we can clearly see the exponential decay characteristic of this geometric sequence. The graph descends rapidly at first and then gradually approaches the x-axis. This is a visual representation of the common ratio (2/3) being less than 1. The closer the common ratio is to 0, the faster the decay. The graph provides valuable insights into the long-term behavior of the sequence. We can see that the terms are getting smaller and smaller, approaching zero as x increases. This is because each term is a fraction (2/3) of the previous term, so the values are diminishing with each step. This behavior is typical of geometric sequences with a common ratio between 0 and 1. We can also use the graph to estimate the value of terms that we haven't calculated directly. For example, we can visually estimate the value of f(5) by looking at the point on the graph that corresponds to x = 5. The graph also highlights the discrete nature of the sequence. The points are distinct and do not form a continuous line. This is because the sequence is only defined for integer values of x. If we were to draw a continuous line through the points, it would not accurately represent the sequence. The graph also helps us to visualize the concept of a limit. In this case, the limit of the sequence as x approaches infinity is 0. This means that the terms of the sequence get arbitrarily close to 0 as x becomes very large. The graph shows this by approaching the x-axis, which represents the line y = 0.

Importance of Initial Value

The initial value, 108, plays a crucial role in determining the position of the graph. If the initial value were different, the entire graph would be shifted vertically. For instance, if the initial value were 54 (half of 108), all the points on the graph would be halved, effectively compressing the graph towards the x-axis. The initial value sets the starting point for the sequence and influences the magnitude of all subsequent terms. A larger initial value will result in larger terms, while a smaller initial value will result in smaller terms. However, the overall shape of the graph – the exponential decay – will remain the same as long as the common ratio remains constant. The initial value also affects the y-intercept of the graph. The y-intercept is the point where the graph intersects the y-axis, which corresponds to the value of the sequence when x = 0. In this case, the y-intercept is (0, 108), which directly reflects the initial value. Understanding the impact of the initial value is essential for analyzing and comparing different geometric sequences. By changing the initial value, we can scale the sequence up or down without altering its fundamental behavior.

Impact of the Common Ratio

Besides the initial value, the common ratio (2/3) is the other key determinant of the sequence's graph. As discussed, it dictates the rate of decay. A smaller common ratio (closer to 0) would result in a steeper descent, while a common ratio closer to 1 would produce a more gradual decline. If the common ratio were greater than 1, the sequence would exhibit exponential growth, and the graph would ascend instead of descend. For example, if the common ratio were 3/2 instead of 2/3, the sequence would increase with each term, and the graph would curve upwards away from the x-axis. The common ratio also affects the long-term behavior of the sequence. If the common ratio is between -1 and 1, the sequence will converge to 0 as x approaches infinity. If the common ratio is greater than 1 or less than -1, the sequence will diverge, meaning that the terms will become infinitely large (in absolute value). If the common ratio is exactly 1, the sequence will be constant, and the graph will be a horizontal line. If the common ratio is exactly -1, the sequence will oscillate between two values, and the graph will alternate between two points. Understanding the impact of the common ratio is crucial for predicting the behavior of a geometric sequence. It allows us to determine whether the sequence will grow, decay, converge, or diverge.

Conclusion

Graphing the sequence defined by f(x+1) = (2/3)f(x) with an initial value of 108 provides a powerful visual tool for understanding the behavior of geometric sequences. By calculating terms, plotting points, and analyzing the resulting graph, we gain valuable insights into the concepts of exponential decay, common ratios, and initial values. This understanding is fundamental for further exploration of sequences, functions, and their applications in various mathematical and real-world contexts. The process of graphing this sequence reinforces key mathematical skills, such as calculating terms, plotting points, and interpreting visual representations. These skills are essential for mathematical problem-solving and analysis in a wide range of contexts. Furthermore, this example provides a concrete illustration of the relationship between a recursive definition and the graphical representation of a sequence. This connection is crucial for developing a deeper understanding of mathematical concepts and their applications. By mastering these concepts, students can confidently tackle more complex problems involving sequences and functions.