Which Functions Are Linear?Check All That Apply.1. $x - Y = 6$2. $y = X^3 - 12$3. $y = 5x$4. $x - 12 = 3y$5. $y = 2x^2 - X$

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Linear Functions: Understanding the Basics

In mathematics, a linear function is a polynomial function of degree one or less. It 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. Linear functions are used to model real-world situations, such as the cost of goods, the distance traveled by an object, and the temperature of a system.

What are Linear Functions?

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, while the y-intercept (b) represents the point at which the function intersects the y-axis.

Examples of Linear Functions

Some examples of linear functions include:

  • y = 2x + 3
  • y = -x + 2
  • y = 3x - 1

Which Functions are Linear?

Now, let's check which of the given functions are linear.

1. x−y=6x - y = 6

To determine if this function is linear, we need to rewrite it in the form of y = mx + b. We can do this by isolating y on one side of the equation.

x - y = 6
-y = -x + 6
y = x - 6

As we can see, this function can be written in the form of y = mx + b, where m = 1 and b = -6. Therefore, this function is linear.

2. y=x3−12y = x^3 - 12

This function is not linear because it is a polynomial function of degree three. It cannot be written in the form of y = mx + b.

3. y=5xy = 5x

This function is linear because it can be written in the form of y = mx + b, where m = 5 and b = 0.

4. x−12=3yx - 12 = 3y

To determine if this function is linear, we need to rewrite it in the form of y = mx + b. We can do this by isolating y on one side of the equation.

x - 12 = 3y
3y = x - 12
y = (1/3)x - 4

As we can see, this function can be written in the form of y = mx + b, where m = 1/3 and b = -4. Therefore, this function is linear.

5. y=2x2−xy = 2x^2 - x

This function is not linear because it is a polynomial function of degree two. It cannot be written in the form of y = mx + b.

Conclusion

In conclusion, the linear functions from the given list are:

  • x−y=6x - y = 6
  • y=5xy = 5x
  • x−12=3yx - 12 = 3y

These functions can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. The other functions are not linear because they are polynomial functions of degree two or three.

Real-World Applications of Linear Functions

Linear functions have many real-world applications, such as:

  • Cost of Goods: The cost of goods is a linear function that can be used to determine the total cost of a product.
  • Distance Traveled: The distance traveled by an object is a linear function that can be used to determine the total distance traveled.
  • Temperature: The temperature of a system is a linear function that can be used to determine the temperature at a given time.

Tips for Identifying Linear Functions

To identify linear functions, you can use the following tips:

  • Check the degree of the polynomial: If the polynomial is of degree one or less, it is a linear function.
  • Check if the function can be written in the form of y = mx + b: If the function can be written in this form, it is a linear function.
  • Check if the function has a constant slope: If the function has a constant slope, it is a linear function.

Conclusion

In conclusion, linear functions are an important concept in mathematics that have many real-world applications. By understanding the basics of linear functions, you can identify which functions are linear and which are not.
Linear Functions: Q&A

In our previous article, we discussed the basics of linear functions and identified which functions are linear. In this article, we will answer some frequently asked questions about linear functions.

Q: What is the difference between a linear function and a non-linear function?

A: A linear function is a polynomial function of degree one or less, while a non-linear function is a polynomial function of degree two or more.

Q: How do I determine if a function is linear?

A: To determine if a function is linear, you can check if it can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. If the function can be written in this form, it is a linear function.

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 is represented by the coefficient of the x-term in the function.

Q: What is the y-intercept of a linear function?

A: The y-intercept of a linear function is the point at which the function intersects the y-axis. It is represented by the constant term in the function.

Q: Can a linear function have a negative slope?

A: Yes, a linear function can have a negative slope. A negative slope indicates that the function is decreasing as x increases.

Q: Can a linear function have a zero slope?

A: Yes, a linear function can have a zero slope. A zero slope indicates that the function is a horizontal line.

Q: Can a linear function have a fractional slope?

A: Yes, a linear function can have a fractional slope. A fractional slope indicates that the function is increasing or decreasing at a rate that is not a whole number.

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

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

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

A: Yes, a linear function can have a fractional y-intercept. A fractional y-intercept indicates that the function intersects the y-axis at a point that is not a whole number.

Q: Can a linear function be a constant function?

A: Yes, a linear function can be a constant function. A constant function is a function that has a slope of zero and a y-intercept that is a constant value.

Q: Can a linear function be a linear equation?

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

Q: Can a linear function be a linear inequality?

A: Yes, a linear function can be a linear inequality. A linear inequality is an inequality that can be written in the form of y < mx + b or y > mx + b, where m is the slope and b is the y-intercept.

Conclusion

In conclusion, linear functions are an important concept in mathematics that have many real-world applications. By understanding the basics of linear functions, you can identify which functions are linear and which are not. We hope that this Q&A article has helped to clarify any questions you may have had about linear functions.

Real-World Applications of Linear Functions

Linear functions have many real-world applications, such as:

  • Cost of Goods: The cost of goods is a linear function that can be used to determine the total cost of a product.
  • Distance Traveled: The distance traveled by an object is a linear function that can be used to determine the total distance traveled.
  • Temperature: The temperature of a system is a linear function that can be used to determine the temperature at a given time.

Tips for Working with Linear Functions

To work with linear functions, you can use the following tips:

  • Check the degree of the polynomial: If the polynomial is of degree one or less, it is a linear function.
  • Check if the function can be written in the form of y = mx + b: If the function can be written in this form, it is a linear function.
  • Check if the function has a constant slope: If the function has a constant slope, it is a linear function.

Conclusion

In conclusion, linear functions are an important concept in mathematics that have many real-world applications. By understanding the basics of linear functions, you can identify which functions are linear and which are not. We hope that this Q&A article has helped to clarify any questions you may have had about linear functions.