Which Table Represents A Linear Function?A.${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 1 & 5 \ \hline 2 & 9 \ \hline 3 & 5 \ \hline 4 & 9 \ \hline \end{tabular} } B . B. B . [ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 1 &
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. A linear function represents a straight line on a graph. In this article, we will explore which table represents a linear function.
What is a Linear Function?
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 slope (m) represents the rate of change of the function, and the y-intercept (b) represents the point where the function intersects the y-axis.
Characteristics of a Linear Function
A linear function has the following characteristics:
- It is a straight line on a graph.
- It can be written in the form of y = mx + b.
- The slope (m) is constant.
- The y-intercept (b) is constant.
Analyzing Table A
Let's analyze Table A to determine if it represents a linear function.
| 1 | 5 |
| 2 | 9 |
| 3 | 5 |
| 4 | 9 |
Checking for a Constant Slope
To determine if Table A represents a linear function, we need to check if the slope is constant. We can calculate the slope by dividing the change in y by the change in x.
| Change in y | Change in x | Slope | ||
|---|---|---|---|---|
| 1 | 5 | -4 | 1 | -4 |
| 2 | 9 | 4 | 1 | 4 |
| 3 | 5 | -4 | 1 | -4 |
| 4 | 9 | 4 | 1 | 4 |
As we can see, the slope is not constant. The slope is -4 when x = 1 and x = 3, and the slope is 4 when x = 2 and x = 4. Therefore, Table A does not represent a linear function.
Analyzing Table B
Let's analyze Table B to determine if it represents a linear function.
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
Checking for a Constant Slope
To determine if Table B represents a linear function, we need to check if the slope is constant. We can calculate the slope by dividing the change in y by the change in x.
| Change in y | Change in x | Slope | ||
|---|---|---|---|---|
| 1 | 3 | 2 | 1 | 2 |
| 2 | 5 | 2 | 1 | 2 |
| 3 | 7 | 2 | 1 | 2 |
| 4 | 9 | 2 | 1 | 2 |
As we can see, the slope is constant. The slope is 2 for all values of x. Therefore, Table B represents a linear function.
Conclusion
In conclusion, Table B represents a linear function because it has a constant slope. Table A does not represent a linear function because it does not have a constant slope.
Key Takeaways
- A linear function is a function that can be written in the form of y = mx + b.
- A linear function has a constant slope and a constant y-intercept.
- Table B represents a linear function because it has a constant slope.
- Table A does not represent a linear function because it does not have a constant slope.
Final Thoughts
Introduction
In our previous article, we explored which table represents a linear function. In this article, we will answer some frequently asked questions about linear functions.
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. The slope (m) represents the rate of change of the function, and the y-intercept (b) represents the point where the function intersects the y-axis.
Q: What are the characteristics of a linear function?
A: A linear function has the following characteristics:
- It is a straight line on a graph.
- It can be written in the form of y = mx + b.
- The slope (m) is constant.
- The y-intercept (b) is constant.
Q: How do I determine if a function is linear?
A: To determine if a function is linear, you need to check if the slope is constant. You can calculate the slope by dividing the change in y by the change in x.
Q: What is the slope of a linear function?
A: The slope of a linear function is the rate of change of the function. It represents how much the function changes for a one-unit change in the input variable.
Q: What is the y-intercept of a linear function?
A: The y-intercept of a linear function is the point where the function intersects the y-axis. It represents the value of the function when the input variable is zero.
Q: Can a linear function have a negative slope?
A: Yes, a linear function can have a negative slope. A negative slope represents a downward trend in the function.
Q: Can a linear function have a zero slope?
A: Yes, a linear function can have a zero slope. A zero slope represents a horizontal line on the graph.
Q: Can a linear function have a fractional slope?
A: Yes, a linear function can have a fractional slope. A fractional slope represents a steeper or less steep trend in the function.
Q: Can a linear function have a slope of 1?
A: Yes, a linear function can have a slope of 1. A slope of 1 represents a straight line with a 45-degree angle.
Q: Can a linear function have a slope of -1?
A: Yes, a linear function can have a slope of -1. A slope of -1 represents a straight line with a -45-degree angle.
Q: Can a linear function have a slope of 0?
A: Yes, a linear function can have a slope of 0. A slope of 0 represents a horizontal line on the graph.
Q: Can a linear function have a slope of infinity?
A: No, a linear function cannot have a slope of infinity. A slope of infinity represents a vertical line on the graph, which is not a linear function.
Conclusion
In conclusion, linear functions are an important concept in mathematics. They can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. A linear function has a constant slope and a constant y-intercept. We hope this Q&A article has helped you understand linear functions better.
Key Takeaways
- A linear function is a function that can be written in the form of y = mx + b.
- A linear function has a constant slope and a constant y-intercept.
- The slope of a linear function represents the rate of change of the function.
- The y-intercept of a linear function represents the point where the function intersects the y-axis.
- A linear function can have a negative slope, a zero slope, a fractional slope, a slope of 1, a slope of -1, or a slope of 0.
- A linear function cannot have a slope of infinity.