Which Function Is Linear?A. $f(x) = 8x^2 + 2$B. $f(x) = \frac{8}{x} + 2$C. $f(x) = 8x^2 - 2$D. $f(x) = \frac{x}{8} - 2$

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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. Linear functions are characterized by their ability to be graphed as a straight line on a coordinate plane. In this article, we will explore four different functions and determine which one is linear.

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, while the y-intercept (b) represents the point at which the function intersects the y-axis. Linear functions are characterized by their ability to be graphed as a straight line on a coordinate plane.

Function A: f(x)=8x2+2f(x) = 8x^2 + 2

Function A is a quadratic function, which means it is not linear. A quadratic function can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants. In this case, the function is f(x)=8x2+2f(x) = 8x^2 + 2, which means it is a quadratic function with a leading coefficient of 8. Since it is a quadratic function, it is not linear.

Function B: f(x)=8x+2f(x) = \frac{8}{x} + 2

Function B is a rational function, which means it is not linear. A rational function is a function that can be written in the form of y = f(x)/g(x), where f(x) and g(x) are polynomials. In this case, the function is f(x)=8x+2f(x) = \frac{8}{x} + 2, which means it is a rational function with a numerator of 8 and a denominator of x. Since it is a rational function, it is not linear.

Function C: f(x)=8x2โˆ’2f(x) = 8x^2 - 2

Function C is a quadratic function, which means it is not linear. A quadratic function can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants. In this case, the function is f(x)=8x2โˆ’2f(x) = 8x^2 - 2, which means it is a quadratic function with a leading coefficient of 8. Since it is a quadratic function, it is not linear.

Function D: f(x)=x8โˆ’2f(x) = \frac{x}{8} - 2

Function D is a linear function. A linear function can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. In this case, the function is f(x)=x8โˆ’2f(x) = \frac{x}{8} - 2, which means it is a linear function with a slope of 1/8 and a y-intercept of -2. Since it is a linear function, it can be graphed as a straight line on a coordinate plane.

Conclusion

In conclusion, the correct answer is D. f(x)=x8โˆ’2f(x) = \frac{x}{8} - 2 is the only linear function among the four options. The other three functions are quadratic or rational functions, which means they are not linear.

Key Takeaways

  • 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 quadratic function is a function that can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants.
  • A rational function is a function that can be written in the form of y = f(x)/g(x), where f(x) and g(x) are polynomials.
  • A linear function can be graphed as a straight line on a coordinate plane.

Final Answer

In our previous article, we discussed the concept of linear functions and identified the correct linear function among four options. In this article, we will provide a Q&A guide to help you better understand linear functions and how to identify them.

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. Linear functions are characterized by their ability to be graphed as a straight line on a coordinate plane.

Q: What are the characteristics of a linear function?

A: A linear function has the following characteristics:

  • It can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.
  • It is a straight line on a coordinate plane.
  • It has a constant rate of change, which is represented by the slope (m).
  • It has a fixed y-intercept, which is represented by the value of b.

Q: How do I identify a linear function?

A: To identify a linear function, you can follow these steps:

  1. Check if the function can be written in the form of y = mx + b.
  2. Check if the function is a straight line on a coordinate plane.
  3. Check if the function has a constant rate of change, which is represented by the slope (m).
  4. Check if the function has a fixed y-intercept, which is represented by the value of b.

Q: What are some examples of linear functions?

A: Some examples of linear functions include:

  • f(x) = 2x + 3
  • f(x) = -x + 2
  • f(x) = 3x - 1
  • f(x) = x/2 + 1

Q: What are some examples of non-linear functions?

A: Some examples of non-linear functions include:

  • f(x) = x^2 + 2
  • f(x) = 1/x + 2
  • f(x) = x^3 - 2
  • f(x) = 2x^2 - 3

Q: Why are linear functions important?

A: Linear functions are important because they can be used to model real-world situations, such as the cost of producing a product, the temperature of a room, or the distance traveled by an object. They are also used in many fields, such as physics, engineering, and economics.

Q: How do I graph a linear function?

A: To graph a linear function, you can follow these steps:

  1. Identify the slope (m) and y-intercept (b) of the function.
  2. Plot the y-intercept (b) on the coordinate plane.
  3. Use the slope (m) to determine the direction and steepness of the line.
  4. Plot additional points on the line using the slope (m) and y-intercept (b).

Q: What are some common mistakes to avoid when working with linear functions?

A: Some common mistakes to avoid when working with linear functions include:

  • Assuming that a function is linear when it is not.
  • Failing to check if a function can be written in the form of y = mx + b.
  • Failing to check if a function is a straight line on a coordinate plane.
  • Failing to check if a function has a constant rate of change, which is represented by the slope (m).

Conclusion

In conclusion, linear functions are an important concept in mathematics and are used to model real-world situations. By understanding the characteristics of linear functions and how to identify them, you can better understand and work with linear functions. Remember to avoid common mistakes when working with linear functions, and always check if a function can be written in the form of y = mx + b.

Key Takeaways

  • 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 is a straight line on a coordinate plane.
  • A linear function has a constant rate of change, which is represented by the slope (m).
  • A linear function has a fixed y-intercept, which is represented by the value of b.

Final Answer

The final answer is that linear functions are an important concept in mathematics and are used to model real-world situations. By understanding the characteristics of linear functions and how to identify them, you can better understand and work with linear functions.