Select The Correct Answer.Which Table Represents The Increasing Linear Function With The Greatest Unit Rate?A. \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline 2 & 16 \ \hline 5 & 10

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Introduction

In mathematics, a linear function is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. The unit rate of a linear function is the ratio of the change in the output (y) to the change in the input (x). In this article, we will explore how to identify the increasing linear function with the greatest unit rate from a given set of tables.

Understanding Unit Rate

The unit rate of a linear function is a measure of how much the output changes when the input changes by one unit. It is calculated by dividing the change in the output (y) by the change in the input (x). In other words, it is the slope of the linear function.

Analyzing the Tables

We are given two tables to analyze:

Table 1

x y
2 16
5 10

Table 2

x y
2 16
5 25

Calculating the Unit Rate

To calculate the unit rate, we need to find the change in the output (y) and the change in the input (x) for each table.

Table 1

  • Change in output (y) = 10 - 16 = -6
  • Change in input (x) = 5 - 2 = 3
  • Unit rate = -6/3 = -2

Table 2

  • Change in output (y) = 25 - 16 = 9
  • Change in input (x) = 5 - 2 = 3
  • Unit rate = 9/3 = 3

Comparing the Unit Rates

Now that we have calculated the unit rates for both tables, we can compare them to determine which one has the greatest unit rate.

  • Table 1 has a unit rate of -2
  • Table 2 has a unit rate of 3

Since 3 is greater than -2, Table 2 has the greatest unit rate.

Conclusion

In conclusion, to select the correct answer, we need to calculate the unit rate for each table and compare them to determine which one has the greatest unit rate. In this case, Table 2 has the greatest unit rate.

Discussion

This problem requires the application of mathematical concepts, such as linear functions and unit rates. It also requires critical thinking and analysis to compare the unit rates and select the correct answer.

Key Takeaways

  • A linear function is a function that can be written in the form of y = mx + b.
  • The unit rate of a linear function is the ratio of the change in the output (y) to the change in the input (x).
  • To calculate the unit rate, we need to find the change in the output (y) and the change in the input (x).
  • To compare the unit rates, we need to divide the change in the output (y) by the change in the input (x).

References

Q: What is a linear function?

A: A linear function is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the unit rate of a linear function?

A: The unit rate of a linear function is the ratio of the change in the output (y) to the change in the input (x).

Q: How do I calculate the unit rate of a linear function?

A: To calculate the unit rate, you need to find the change in the output (y) and the change in the input (x). Then, divide the change in the output (y) by the change in the input (x).

Q: What is the difference between a positive and negative unit rate?

A: A positive unit rate indicates that the output increases as the input increases. A negative unit rate indicates that the output decreases as the input increases.

Q: How do I compare the unit rates of two linear functions?

A: To compare the unit rates, you need to divide the change in the output (y) by the change in the input (x) for each function. The function with the greater unit rate has a greater rate of change.

Q: What is the significance of the unit rate in real-world applications?

A: The unit rate is significant in real-world applications because it helps to understand the rate of change of a function. It can be used to make predictions, model real-world phenomena, and make informed decisions.

Q: Can you provide an example of a real-world application of the unit rate?

A: Yes, the unit rate is used in finance to calculate the interest rate on a loan. For example, if a loan has a unit rate of 0.05, it means that for every dollar borrowed, the borrower will pay 5 cents in interest.

Q: How do I determine which linear function has the greatest unit rate?

A: To determine which linear function has the greatest unit rate, you need to calculate the unit rate for each function and compare them. The function with the greatest unit rate has the greatest rate of change.

Q: Can you provide an example of a problem that requires determining the greatest unit rate?

A: Yes, consider the following problem:

Which table represents the increasing linear function with the greatest unit rate?

x y
2 16
5 10
x y
--- ---
2 16
5 25

To solve this problem, you need to calculate the unit rate for each table and compare them. The table with the greatest unit rate represents the increasing linear function with the greatest unit rate.

Q: How do I use the unit rate to make predictions?

A: The unit rate can be used to make predictions by extrapolating the rate of change of a function. For example, if a function has a unit rate of 0.05, you can predict that the output will increase by 5 cents for every dollar increase in the input.

Q: Can you provide an example of a problem that requires using the unit rate to make predictions?

A: Yes, consider the following problem:

A company's sales are increasing at a rate of 0.05 per dollar. If the company's sales are currently $100,000, how much will they be in 5 years?

To solve this problem, you need to use the unit rate to make predictions. The unit rate indicates that the sales will increase by 5 cents for every dollar increase in the input. Therefore, in 5 years, the sales will be $100,000 + (5 x $100,000) = $500,000.

Conclusion

In conclusion, the unit rate is a significant concept in mathematics that helps to understand the rate of change of a function. It can be used to make predictions, model real-world phenomena, and make informed decisions. By understanding the unit rate, you can solve problems that require determining the greatest unit rate and make predictions about the behavior of a function.