Several Ordered Pairs From A Continuous Exponential Function Are Shown In The Table.${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 0 & 4 \ \hline 1 & 5 \ \hline 2 & 6.25 \ \hline 3 & 7.8125 \ \hline \end{tabular} }$What Are The

Mar 11, 2025 by ADMIN 238 views

**Analyzing Exponential Functions: A Closer Look at Ordered Pairs**

Introduction

In mathematics, exponential functions are a fundamental concept that plays a crucial role in various fields, including physics, engineering, and economics. These functions describe the relationship between two variables, where one variable is a constant power of the other. In this article, we will delve into the world of exponential functions and explore how to analyze them using ordered pairs.

Understanding Exponential Functions

An exponential function is a mathematical function of the form $y = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is the constant that determines the rate of growth or decay of the function. If $b > 1$ , the function grows exponentially, and if $0 < b < 1$ , the function decays exponentially.

Analyzing Ordered Pairs

Ordered pairs are a way to represent the relationship between two variables in a mathematical function. In the context of exponential functions, ordered pairs can be used to identify the values of $x$ and $y$ that satisfy the function. The table below shows several ordered pairs from a continuous exponential function.

$x$	$y$
0	4
1	5
2	6.25
3	7.8125

Identifying the Exponential Function

To identify the exponential function that corresponds to the ordered pairs, we need to analyze the pattern of the values. Looking at the table, we can see that each value of $y$ is obtained by multiplying the previous value by a constant factor.

$x$	$y$	Factor
0	4	-
1	5	1.25
2	6.25	1.25
3	7.8125	1.25

The factor is 1.25, which is the base of the exponential function. Therefore, the exponential function that corresponds to the ordered pairs is $y = 4(1.25)^x$ .

Verifying the Exponential Function

To verify that the exponential function $y = 4(1.25)^x$ corresponds to the ordered pairs, we can substitute each value of $x$ into the function and calculate the corresponding value of $y$ .

$x$	$y = 4(1.25)^x$
0	4
1	5
2	6.25
3	7.8125

The calculated values of $y$ match the values in the table, which confirms that the exponential function $y = 4(1.25)^x$ corresponds to the ordered pairs.

Conclusion

In this article, we analyzed several ordered pairs from a continuous exponential function and identified the corresponding exponential function. We also verified that the function corresponds to the ordered pairs by substituting each value of $x$ into the function and calculating the corresponding value of $y$ . This demonstrates the importance of analyzing ordered pairs in understanding exponential functions.

Exercises

Find the exponential function that corresponds to the ordered pairs in the table below.

$x$ $y$

0 2

1 3

2 4.5

3 6.75
Verify that the exponential function $y = 2(1.5)^x$ corresponds to the ordered pairs in the table below.

$x$ $y = 2(1.5)^x$

0 2

1 3

2 4.5

3 6.75

$x$	$y$
0	2
1	3
2	4.5
3	6.75

$x$	$y = 2(1.5)^x$
0	2
1	3
2	4.5
3	6.75

Solutions

The exponential function that corresponds to the ordered pairs is $y = 2(1.5)^x$ .

To verify this, we can substitute each value of $x$ into the function and calculate the corresponding value of $y$ .

$x$ $y = 2(1.5)^x$

0 2

1 3

2 4.5

3 6.75

The calculated values of $y$ match the values in the table, which confirms that the exponential function $y = 2(1.5)^x$ corresponds to the ordered pairs.
The exponential function $y = 2(1.5)^x$ corresponds to the ordered pairs in the table.

To verify this, we can substitute each value of $x$ into the function and calculate the corresponding value of $y$ .

$x$ $y = 2(1.5)^x$

0 2

1 3

2 4.5

3 6.75

The calculated values of $y$ match the values in the table, which confirms that the exponential function $y = 2(1.5)^x$ corresponds to the ordered pairs.
Q&A: Exponential Functions and Ordered Pairs =============================================

$x$	$y = 2(1.5)^x$
0	2
1	3
2	4.5
3	6.75

$x$	$y = 2(1.5)^x$
0	2
1	3
2	4.5
3	6.75

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form $y = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable. The base $b$ is the constant that determines the rate of growth or decay of the function.

Q: What is the difference between a linear function and an exponential function?

A: A linear function is a function of the form $y = mx + b$ , where $m$ and $b$ are constants. An exponential function, on the other hand, is a function of the form $y = ab^x$ , where $a$ and $b$ are constants. The key difference between the two is that a linear function has a constant rate of change, while an exponential function has a rate of change that increases or decreases exponentially.

Q: How do I identify the exponential function that corresponds to a set of ordered pairs?

A: To identify the exponential function that corresponds to a set of ordered pairs, you need to analyze the pattern of the values. Look for a constant factor that is multiplied by the previous value to obtain the next value. This factor is the base of the exponential function.

Q: How do I verify that an exponential function corresponds to a set of ordered pairs?

A: To verify that an exponential function corresponds to a set of ordered pairs, you need to substitute each value of $x$ into the function and calculate the corresponding value of $y$ . If the calculated values of $y$ match the values in the table, then the exponential function corresponds to the ordered pairs.

Q: What is the significance of the base in an exponential function?

A: The base in an exponential function determines the rate of growth or decay of the function. If the base is greater than 1, the function grows exponentially. If the base is between 0 and 1, the function decays exponentially.

Q: Can an exponential function have a negative base?

A: Yes, an exponential function can have a negative base. However, the function will decay exponentially if the base is negative.

Q: Can an exponential function have a base of 1?

A: Yes, an exponential function can have a base of 1. In this case, the function is a linear function, and the rate of change is constant.

Q: Can an exponential function have a base of 0?

A: No, an exponential function cannot have a base of 0. This is because division by zero is undefined.

Q: Can an exponential function have a negative exponent?

A: Yes, an exponential function can have a negative exponent. In this case, the function is a reciprocal function, and the rate of change is inversely proportional to the base.

Q: Can an exponential function have a fractional exponent?

A: Yes, an exponential function can have a fractional exponent. In this case, the function is a power function, and the rate of change is proportional to the base raised to the power of the exponent.

Q: Can an exponential function have a complex exponent?

A: Yes, an exponential function can have a complex exponent. In this case, the function is a complex exponential function, and the rate of change is proportional to the base raised to the power of the complex exponent.

Conclusion

In this article, we have answered some common questions about exponential functions and ordered pairs. We have discussed the definition of an exponential function, the difference between a linear function and an exponential function, and how to identify and verify an exponential function that corresponds to a set of ordered pairs. We have also discussed the significance of the base in an exponential function and some special cases, such as negative bases, bases of 1, and fractional exponents.