Determine whether each function is linear or not linear

Oct 13, 2025 by windailys.com 112 views

Determine whether each function is linear or not linear

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 has a constant rate of change, and its graph is a straight line. On the other hand, a non-linear function is a function that does not have a constant rate of change, and its graph is not a straight line. In this article, we will determine whether each of the given functions is linear or not linear.

The first function is given by the set of points $\{(-2,5),(1,4),(4,1),(7,-1)\}$ . To determine whether this function is linear or not linear, we need to calculate the slope between each pair of points.

Calculating the Slope

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

Let's calculate the slope between each pair of points:

Between $(-2,5)$ and $(1,4)$ : $$m = \frac{4 - 5}{1 - (-2)} = \frac{-1}{3}$$
Between $(1,4)$ and $(4,1)$ : $$m = \frac{1 - 4}{4 - 1} = \frac{-3}{3} = -1$$
Between $(4,1)$ and $(7,-1)$ : $$m = \frac{-1 - 1}{7 - 4} = \frac{-2}{3}$$

Analyzing the Results

We can see that the slopes between each pair of points are not the same. In fact, the slopes are different for each pair of points. This suggests that the function is not linear.

Based on the calculations, we can conclude that the first function is not linear.

The second function is given by the set of points $\{(4,0),(6,0),(8,0),(10,0)\}$ . To determine whether this function is linear or not linear, we need to calculate the slope between each pair of points.

Calculating the Slope

Let's calculate the slope between each pair of points:

Between $(4,0)$ and $(6,0)$ : $$m = \frac{0 - 0}{6 - 4} = 0$$
Between $(6,0)$ and $(8,0)$ : $$m = \frac{0 - 0}{8 - 6} = 0$$
Between $(8,0)$ and $(10,0)$ : $$m = \frac{0 - 0}{10 - 8} = 0$$

Analyzing the Results

We can see that the slopes between each pair of points are the same, and they are all equal to 0. This suggests that the function is linear.

Based on the calculations, we can conclude that the second function is linear.

In this article, we determined whether each of the given functions is linear or not linear. We used the concept of slope to analyze the functions and concluded that the first function is not linear, while the second function is linear.

Key Takeaways

A linear function has a constant rate of change, and its graph is a straight line.
A non-linear function does not have a constant rate of change, and its graph is not a straight line.
The slope between two points can be used to determine whether a function is linear or not linear.

Real-World Applications

Understanding whether a function is linear or not linear has many real-world applications. For example, in economics, a linear function can be used to model the relationship between the price of a good and the quantity demanded. In physics, a linear function can be used to model the motion of an object under constant acceleration.

In conclusion, determining whether a function is linear or not linear is an important concept in mathematics. By using the concept of slope, we can analyze functions and conclude whether they are linear or not linear. This knowledge has many real-world applications and is essential for understanding many mathematical concepts.

[1] Khan Academy. (n.d.). Linear Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4-linear-functions
[2] Math Open Reference. (n.d.). Linear Functions. Retrieved from https://www.mathopenref.com/linfunc.html

Linear function: A function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.
Non-linear function: A function that does not have a constant rate of change, and its graph is not a straight line.
Slope: The rate of change of a function, calculated as the difference in y-values divided by the difference in x-values.
Determine whether each function is linear or not linear: Q&A

In our previous article, we determined whether each of the given functions is linear or not linear. We used the concept of slope to analyze the functions and concluded that the first function is not linear, while the second function is linear. In this article, we will answer some frequently asked questions related to linear and non-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. A linear function has a constant rate of change, and its graph is a straight line.

Q: What is a non-linear function?

A: A non-linear function is a function that does not have a constant rate of change, and its graph is not a straight line. Non-linear functions can be quadratic, cubic, or any other type of function that does not have a constant rate of change.

Q: How do I determine whether a function is linear or not linear?

A: To determine whether a function is linear or not linear, you can use the concept of slope. Calculate the slope between each pair of points on the function, and if the slopes are the same, then the function is linear. If the slopes are different, then the function is not linear.

Q: What are some examples of linear functions?

A: Some examples of linear functions include:

y = 2x + 3
y = -x + 2
y = 4x - 1

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

A: Some examples of non-linear functions include:

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

Q: Can a non-linear function be a quadratic function?

A: Yes, a non-linear function can be a quadratic function. 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. Quadratic functions are non-linear because they do not have a constant rate of change.

Q: Can a linear function be a polynomial function?

A: Yes, a linear function can be a polynomial function. A polynomial function is a function that can be written in the form of y = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants. A linear function is a polynomial function of degree 1.

Q: What are some real-world applications of linear and non-linear functions?

A: Linear and non-linear functions have many real-world applications. Some examples include:

Modeling the relationship between the price of a good and the quantity demanded in economics
Modeling the motion of an object under constant acceleration in physics
Modeling the growth of a population in biology
Modeling the spread of a disease in epidemiology

[1] Khan Academy. (n.d.). Linear Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4-linear-functions
[2] Math Open Reference. (n.d.). Linear Functions. Retrieved from https://www.mathopenref.com/linfunc.html

Linear function: A function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.
Non-linear function: A function that does not have a constant rate of change, and its graph is not a straight line.
Slope: The rate of change of a function, calculated as the difference in y-values divided by the difference in x-values.
Quadratic function: A function that can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants.
Polynomial function: A function that can be written in the form of y = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.