Which Is The Graph Of The Sequence Defined By The Function $f(x+1)=\frac{2}{3} F(x$\] If The Initial Value Of The Sequence Is Given?

Introduction

In mathematics, sequences and functions are fundamental concepts that help us understand various mathematical relationships and patterns. A sequence is a list of numbers or values that follow a specific order, while a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). In this article, we will explore the graph of a sequence defined by a function, specifically the function $f(x+1)=\frac{2}{3} f(x)$, and discuss how to determine the graph of the sequence given its initial value.

The Function $f(x+1)=\frac{2}{3} f(x)$

The given function is a recursive function, meaning that each term in the sequence is defined in terms of the previous term. The function states that the value of $f(x+1)$ is equal to $\frac{2}{3}$ times the value of $f(x)$. This means that each term in the sequence is obtained by multiplying the previous term by $\frac{2}{3}$.

Understanding the Graph of the Sequence

To understand the graph of the sequence, we need to visualize the values of the sequence as a function of $x$. Since the function is recursive, we can start by finding the first few terms of the sequence. Let's assume that the initial value of the sequence is $f(0) = a$, where $a$ is a constant.

Finding the First Few Terms of the Sequence

Using the recursive function, we can find the first few terms of the sequence as follows:

• $f(1) = \frac{2}{3} f(0) = \frac{2}{3} a$
• $f(2) = \frac{2}{3} f(1) = \frac{2}{3} \left(\frac{2}{3} a\right) = \left(\frac{2}{3}\right)^2 a$
• $f(3) = \frac{2}{3} f(2) = \frac{2}{3} \left(\frac{2}{3}\right)^2 a = \left(\frac{2}{3}\right)^3 a$

Identifying the Pattern in the Sequence

From the first few terms of the sequence, we can see a pattern emerging. Each term is obtained by multiplying the previous term by $\frac{2}{3}$. This means that the sequence is a geometric sequence with a common ratio of $\frac{2}{3}$.

The General Formula for the Sequence

Using the pattern identified in the previous section, we can write a general formula for the sequence as follows:

$f(x) = \left(\frac{2}{3}\right)^x a$

This formula states that the value of $f(x)$ is equal to $a$ multiplied by $\left(\frac{2}{3}\right)^x$, where $x$ is the term number.

Graphing the Sequence

To graph the sequence, we can plot the values of $f(x)$ against $x$. Since the sequence is a geometric sequence, the graph will be a straight line with a slope of $\log_{\frac{2}{3}} \left(\frac{2}{3}\right) = 1$.

The Initial Value of the Sequence

The initial value of the sequence, $f(0) = a$, is a critical parameter that determines the graph of the sequence. If we know the initial value of the sequence, we can determine the graph of the sequence using the general formula.

Conclusion

In this article, we explored the graph of a sequence defined by the function $f(x+1)=\frac{2}{3} f(x)$. We found that the sequence is a geometric sequence with a common ratio of $\frac{2}{3}$ and derived a general formula for the sequence. We also discussed how to graph the sequence and the importance of the initial value of the sequence in determining the graph.

Example 1: Finding the Graph of the Sequence

Suppose we are given the initial value of the sequence, $f(0) = 10$. Using the general formula, we can find the graph of the sequence as follows:

$f(x) = \left(\frac{2}{3}\right)^x 10$

To graph the sequence, we can plot the values of $f(x)$ against $x$. The graph will be a straight line with a slope of $1$.

Example 2: Finding the Initial Value of the Sequence

Suppose we are given the graph of the sequence and we want to find the initial value of the sequence. Let's assume that the graph of the sequence is a straight line with a slope of $1$. Using the general formula, we can write:

$f(x) = \left(\frac{2}{3}\right)^x a$

Since the graph is a straight line with a slope of $1$, we know that $a = 10$. Therefore, the initial value of the sequence is $f(0) = 10$.

Conclusion

In this article, we explored the graph of a sequence defined by the function $f(x+1)=\frac{2}{3} f(x)$. We found that the sequence is a geometric sequence with a common ratio of $\frac{2}{3}$ and derived a general formula for the sequence. We also discussed how to graph the sequence and the importance of the initial value of the sequence in determining the graph.

Q: What is the graph of a sequence defined by a function?

A: The graph of a sequence defined by a function is a visual representation of the values of the sequence as a function of the term number. In this article, we explored the graph of a sequence defined by the function $f(x+1)=\frac{2}{3} f(x)$.

Q: What is a recursive function?

A: A recursive function is a function that is defined in terms of itself. In the case of the function $f(x+1)=\frac{2}{3} f(x)$, each term in the sequence is defined in terms of the previous term.

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. In the case of the sequence defined by the function $f(x+1)=\frac{2}{3} f(x)$, the common ratio is $\frac{2}{3}$.

Q: How do I determine the graph of a sequence given its initial value?

A: To determine the graph of a sequence given its initial value, you can use the general formula for the sequence, which is $f(x) = \left(\frac{2}{3}\right)^x a$, where $a$ is the initial value of the sequence.

Q: What is the importance of the initial value of the sequence in determining the graph?

A: The initial value of the sequence is a critical parameter that determines the graph of the sequence. If you know the initial value of the sequence, you can determine the graph of the sequence using the general formula.

Q: How do I find the initial value of the sequence given its graph?

A: To find the initial value of the sequence given its graph, you can use the general formula for the sequence, which is $f(x) = \left(\frac{2}{3}\right)^x a$. Since the graph is a straight line with a slope of $1$, you can solve for $a$ to find the initial value of the sequence.

Q: What is the relationship between the graph of a sequence and its recursive function?

A: The graph of a sequence is a visual representation of the values of the sequence as a function of the term number. The recursive function defines the relationship between each term in the sequence and the previous term.

Q: Can I use the graph of a sequence to determine its recursive function?

A: Yes, you can use the graph of a sequence to determine its recursive function. By analyzing the graph, you can identify the pattern of the sequence and determine the recursive function that defines it.

Q: What are some common applications of the graph of a sequence?

A: The graph of a sequence has many common applications in mathematics, science, and engineering. Some examples include:

• Modeling population growth and decline
• Analyzing the behavior of complex systems
• Optimizing the performance of algorithms and data structures
• Visualizing the results of scientific experiments and simulations

Q: How do I graph a sequence using a calculator or computer?

A: To graph a sequence using a calculator or computer, you can use a graphing calculator or a computer program such as MATLAB or Python. These tools allow you to input the sequence and its recursive function, and then visualize the graph of the sequence.

Q: What are some common mistakes to avoid when graphing a sequence?

A: Some common mistakes to avoid when graphing a sequence include:

• Failing to identify the pattern of the sequence
• Using an incorrect recursive function
• Not considering the initial value of the sequence
• Not using a sufficient number of terms to accurately represent the sequence

Conclusion

In this article, we explored the graph of a sequence defined by a function, specifically the function $f(x+1)=\frac{2}{3} f(x)$. We discussed how to determine the graph of a sequence given its initial value, and how to find the initial value of the sequence given its graph. We also covered some common applications of the graph of a sequence and provided tips for graphing a sequence using a calculator or computer.