Understanding Linear Functions and Unit Rates

Oct 13, 2025 by windailys.com 124 views

Which Table Represents a Linear Function with a Greater Unit Rate?

Understanding Linear Functions and Unit Rates

A linear function is a type of mathematical function that represents a straight line on a graph. It is typically represented in the form of y = mx + b, where m is the slope or unit rate, and b is the y-intercept. The unit rate, also known as the slope, is a measure of the rate at which the output (y) changes in response to a one-unit change in the input (x). In other words, it represents the rate at which the function increases or decreases as the input variable changes.

The Given Function

The given function is y = 200x + 50. This function has a slope of 200, which means that for every one-unit increase in x, the output y increases by 200 units. This is the unit rate of the function.

Analyzing the Options

We are given two tables to analyze and determine which one represents a linear function with a greater unit rate than the given function.

Table A

x	y
1	235
3	595

To determine the unit rate of this function, we need to calculate the slope. We can do this by finding the change in y (Δy) and dividing it by the change in x (Δx).

Δy = 595 - 235 = 360 Δx = 3 - 1 = 2

Now, we can calculate the slope (m) by dividing Δy by Δx:

m = Δy / Δx = 360 / 2 = 180

Table B

x	y
1	250
3	650

To determine the unit rate of this function, we need to calculate the slope. We can do this by finding the change in y (Δy) and dividing it by the change in x (Δx).

Δy = 650 - 250 = 400 Δx = 3 - 1 = 2

Now, we can calculate the slope (m) by dividing Δy by Δx:

m = Δy / Δx = 400 / 2 = 200

Comparing the Unit Rates

Now that we have calculated the unit rates of both functions, we can compare them to determine which one has a greater unit rate.

The unit rate of Table A is 180, while the unit rate of Table B is 200. Since 200 is greater than 180, Table B represents a linear function with a greater unit rate than the given function.

Conclusion

In conclusion, Table B represents a linear function with a greater unit rate than the given function y = 200x + 50. This is because the unit rate of Table B is 200, which is greater than the unit rate of the given function, which is 200.

Key Takeaways

A linear function is a type of mathematical function that represents a straight line on a graph.
The unit rate, also known as the slope, is a measure of the rate at which the output (y) changes in response to a one-unit change in the input (x).
To determine the unit rate of a linear function, we need to calculate the slope by finding the change in y (Δy) and dividing it by the change in x (Δx).
Table B represents a linear function with a greater unit rate than the given function y = 200x + 50.

Final Answer

The final answer is Table B.
Frequently Asked Questions (FAQs) about Linear Functions and Unit Rates

Q: What is a linear function?

A: A linear function is a type of mathematical function that represents a straight line on a graph. It is typically represented in the form of y = mx + b, where m is the slope or unit rate, and b is the y-intercept.

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

A: The unit rate, also known as the slope, is a measure of the rate at which the output (y) changes in response to a one-unit change in the input (x). It represents the rate at which the function increases or decreases as the input variable changes.

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

A: To calculate the unit rate of a linear function, you need to find the change in y (Δy) and divide it by the change in x (Δx). This will give you the slope of the function, which is the unit rate.

Q: What is the difference between the slope and the y-intercept of a linear function?

A: The slope (m) is the rate at which the output (y) changes in response to a one-unit change in the input (x). The y-intercept (b) is the point at which the function intersects the y-axis.

Q: How do I determine which table represents a linear function with a greater unit rate?

A: To determine which table represents a linear function with a greater unit rate, you need to calculate the slope of each function by finding the change in y (Δy) and dividing it by the change in x (Δx). The table with the greater slope represents the linear function with the greater unit rate.

Q: Can a linear function have a negative unit rate?

A: Yes, a linear function can have a negative unit rate. This means that the function decreases as the input variable increases.

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 represents the rate at which a quantity changes in response to a change in another quantity. For example, in finance, the unit rate represents the rate of return on an investment. In physics, the unit rate represents the rate of change of velocity.

Q: Can a linear function have a zero unit rate?

A: Yes, a linear function can have a zero unit rate. This means that the function does not change as the input variable changes.

Q: How do I graph a linear function?

A: To graph a linear function, you need to plot two points on the graph and draw a straight line through them. The slope of the function represents the rate at which the line rises or falls.

Q: Can a linear function have a fractional unit rate?

A: Yes, a linear function can have a fractional unit rate. This means that the function changes at a rate that is a fraction of the input variable.

Q: What is the relationship between the unit rate and the y-intercept of a linear function?

A: The unit rate (m) and the y-intercept (b) are related in that the y-intercept is the point at which the function intersects the y-axis, and the unit rate represents the rate at which the function changes as the input variable changes.

Q: Can a linear function have a negative y-intercept?

A: Yes, a linear function can have a negative y-intercept. This means that the function intersects the y-axis at a point below the origin.

Q: How do I determine the equation of a linear function given its graph?

A: To determine the equation of a linear function given its graph, you need to find the slope (m) and the y-intercept (b) of the function. The equation of the function is then given by y = mx + b.

Q: Can a linear function have a fractional y-intercept?

A: Yes, a linear function can have a fractional y-intercept. This means that the function intersects the y-axis at a point that is a fraction of the y-axis.

Q: What is the significance of the unit rate in economics?

A: The unit rate is significant in economics because it represents the rate of return on an investment. It is used to calculate the interest rate on a loan or the rate of return on an investment.

Q: Can a linear function have a negative slope and a positive y-intercept?

A: Yes, a linear function can have a negative slope and a positive y-intercept. This means that the function decreases as the input variable increases, but it intersects the y-axis at a point above the origin.

Q: How do I determine the unit rate of a linear function given its equation?

A: To determine the unit rate of a linear function given its equation, you need to identify the slope (m) of the function. The slope is the coefficient of the x-term in the equation.

Q: Can a linear function have a fractional slope?

A: Yes, a linear function can have a fractional slope. This means that the function changes at a rate that is a fraction of the input variable.

Q: What is the relationship between the unit rate and the x-intercept of a linear function?

A: The unit rate (m) and the x-intercept (x) are related in that the x-intercept is the point at which the function intersects the x-axis, and the unit rate represents the rate at which the function changes as the input variable changes.

Q: Can a linear function have a negative x-intercept?

A: Yes, a linear function can have a negative x-intercept. This means that the function intersects the x-axis at a point to the left of the origin.

Q: How do I determine the equation of a linear function given its x-intercept and y-intercept?

A: To determine the equation of a linear function given its x-intercept and y-intercept, you need to use the formula y = mx + b, where m is the slope and b is the y-intercept. The x-intercept is the point at which the function intersects the x-axis, and the y-intercept is the point at which the function intersects the y-axis.

Q: Can a linear function have a fractional x-intercept?

A: Yes, a linear function can have a fractional x-intercept. This means that the function intersects the x-axis at a point that is a fraction of the x-axis.

Q: What is the significance of the unit rate in physics?

A: The unit rate is significant in physics because it represents the rate of change of velocity. It is used to calculate the acceleration of an object.

Q: Can a linear function have a negative unit rate and a positive x-intercept?

A: Yes, a linear function can have a negative unit rate and a positive x-intercept. This means that the function decreases as the input variable increases, but it intersects the x-axis at a point to the right of the origin.

Q: How do I determine the unit rate of a linear function given its graph and a point on the graph?

A: To determine the unit rate of a linear function given its graph and a point on the graph, you need to find the slope of the function by drawing a line through the point and the origin. The slope is then given by the ratio of the change in y to the change in x.

Q: Can a linear function have a fractional unit rate and a fractional x-intercept?

A: Yes, a linear function can have a fractional unit rate and a fractional x-intercept. This means that the function changes at a rate that is a fraction of the input variable, and it intersects the x-axis at a point that is a fraction of the x-axis.

Q: What is the relationship between the unit rate and the y-intercept of a linear function in terms of the equation of the function?

A: The unit rate (m) and the y-intercept (b) are related in that the equation of the function is given by y = mx + b, where m is the slope and b is the y-intercept. The unit rate represents the rate at which the function changes as the input variable changes, and the y-intercept represents the point at which the function intersects the y-axis.

Q: Can a linear function have a negative unit rate and a negative y-intercept?

A: Yes, a linear function can have a negative unit rate and a negative y-intercept. This means that the function decreases as the input variable increases, and it intersects the y-axis at a point below the origin.

Q: How do I determine the equation of a linear function given its unit rate and a point on the graph?

A: To determine the equation of a linear function given its unit rate and a point on the graph, you need to use the formula y = mx + b, where m is the slope and b is the y-intercept. The unit rate is given by the slope, and the point on the graph is used to find the y-intercept.

Q: Can a linear function have a fractional unit rate and a fractional y-intercept?

A: Yes, a linear function can have a fractional unit rate and a fractional y-intercept. This means that the function changes at a rate that is a fraction of the input variable, and it intersects the y-axis at a point that is a fraction of the y-axis.

Q: What is the significance of the unit rate in engineering?

A: The unit rate is significant in engineering because it represents the rate of change of a quantity. It is used to calculate the rate of change of a system or a process.

**Q: Can a linear