Identifying the Exponential Function from the Table of Ordered Pairs

Oct 13, 2025 by windailys.com 61 views

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as x and y. These functions are characterized by the fact that the variable y changes at a rate proportional to its current value. In other words, exponential functions describe growth or decay that occurs at a constant rate. The general form of an exponential function is y = ab^x, where a and b are constants, and x is the variable.

Identifying the Exponential Function from the Table of Ordered Pairs

The table of ordered pairs ${(x, y)}$ gives us the following information:

x	y
-1	2
0	10
1	50
2	250

To write an equation for the function, we need to identify the values of a and b in the general form of the exponential function, y = ab^x. We can do this by examining the table of ordered pairs and looking for a pattern.

Finding the Value of a

The value of a is the value of y when x = 0. From the table, we can see that when x = 0, y = 10. Therefore, the value of a is 10.

Finding the Value of b

To find the value of b, we need to examine the relationship between the values of x and y. We can do this by looking at the ratio of y to b^x. From the table, we can see that when x = 1, y = 50, and when x = 2, y = 250. We can use these values to find the ratio of y to b^x.

Calculating the Ratio of y to b^x

When x = 1, y = 50, and when x = 2, y = 250. We can use these values to find the ratio of y to b^x.

When x = 1, we have:

y = 50 b^x = b^1 = b

When x = 2, we have:

y = 250 b^x = b^2

We can divide the two equations to get:

250/50 = (b^2)/b 5 = b

Therefore, the value of b is 5.

Writing the Equation for the Function

Now that we have found the values of a and b, we can write the equation for the function. We have:

a = 10 b = 5

The general form of the exponential function is y = ab^x. Substituting the values of a and b, we get:

y = 10(5^x)

This is the equation for the function.

Verifying the Equation

To verify the equation, we can plug in the values of x and y from the table and check if the equation holds true.

When x = -1, y = 2:

2 = 10(5^(-1)) 2 = 10/5 2 = 2

When x = 0, y = 10:

10 = 10(5^0) 10 = 10

When x = 1, y = 50:

50 = 10(5^1) 50 = 10*5 50 = 50

When x = 2, y = 250:

250 = 10(5^2) 250 = 10*25 250 = 250

The equation holds true for all the values of x and y in the table.

Conclusion

In this article, we have shown how to write an equation for an exponential function given a table of ordered pairs. We have identified the values of a and b in the general form of the exponential function, y = ab^x, and verified the equation by plugging in the values of x and y from the table. The equation for the function is y = 10(5^x).

Understanding Exponential Functions

Q&A: Writing an Equation for the Exponential Function

Q: What is the general form of an exponential function?

A: The general form of an exponential function is y = ab^x, where a and b are constants, and x is the variable.

Q: How do I identify the values of a and b in the general form of the exponential function?

A: To identify the values of a and b, you need to examine the table of ordered pairs and look for a pattern. The value of a is the value of y when x = 0, and the value of b is the ratio of y to b^x.

Q: How do I find the value of a?

A: The value of a is the value of y when x = 0. From the table, you can see that when x = 0, y = 10. Therefore, the value of a is 10.

Q: How do I find the value of b?

A: To find the value of b, you need to examine the relationship between the values of x and y. You can do this by looking at the ratio of y to b^x. From the table, you can see that when x = 1, y = 50, and when x = 2, y = 250. You can use these values to find the ratio of y to b^x.

Q: What is the ratio of y to b^x?

A: When x = 1, y = 50, and when x = 2, y = 250. You can divide the two equations to get:

250/50 = (b^2)/b 5 = b

Therefore, the value of b is 5.

Q: How do I write the equation for the function?

A: Now that you have found the values of a and b, you can write the equation for the function. You have:

a = 10 b = 5

The general form of the exponential function is y = ab^x. Substituting the values of a and b, you get:

y = 10(5^x)

This is the equation for the function.

Q: How do I verify the equation?

A: To verify the equation, you can plug in the values of x and y from the table and check if the equation holds true.

Q: What if I have a table of ordered pairs with negative values of x?

A: If you have a table of ordered pairs with negative values of x, you can still use the same method to find the values of a and b. However, you may need to use a calculator or a computer to evaluate the expressions.

Q: Can I use this method to find the equation for any type of function?

A: No, this method is specifically designed for exponential functions. If you have a table of ordered pairs for a different type of function, you will need to use a different method to find the equation.

Conclusion

In this article, we have provided a Q&A section to help you understand how to write an equation for an exponential function given a table of ordered pairs. We have answered common questions and provided examples to help you understand the process.