The word linear literally means arranged in a straight line. So a linear function can be defined as a function that represents a straight line. In a coordinate plane, a straight line is represented by the linear function. Let us take an example of a linear function to understand it more clearly. The equation y = 7x – 13 is a linear function because it represents a straight line on graph paper. In this article, we will discuss in detail the linear functions and their various concepts along with taking examples to understand the various concepts clearly.
What Do You Mean by a Linear Function?
We can define a linear function as a function that represents a straight line on a coordinate plane. It is of the form y = f(x) = mx + b where m is a real number and b is a real number. In this equation, we have a dependent variable and an independent variable. We have y as the dependent and x as the independent variable. If you notice carefully, you will understand that this equation is similar to the slope-intercept form of a line. Let us take some examples of a linear to understand it more clearly.
- f(x) = y = 5x – 12
- f(x) = y = -6x – 7
- f(x) = y = 2x – 13
- f(x) = y = 13x – 1
Some Real Life Examples of Linear Function
- A taxi service charges a fee of Rs 200 every month and an additional fee of Rs 10 per kilometer. Here, we can represent this scenario in the form of a linear equation y = 10x + 200 where y is the total cost and x represents total kilometers traveled.
- A web series streaming service charges a yearly fee of Rs 500 and an additional fee of Rs 50 for every web series that an individual watches. Here, we can represent this scenario in the form of a linear equation which is y = 50x + 500 where x is the total number of web series watched.
- We use the concept of linear functions in the chapter on linear programming problems. Linear programming problems are used to find optimal solutions.
How Do We Find a Linear Function?
To find out a linear function, we use the point-slope form. To find out a linear function, we use the same steps as we do in the case of finding the equation of a line. Let us understand how to find out a linear function with the help of an example.
Example: Obtain the equation of a linear function that has two points (6, 10) and (8, 12).
Solution: It has given that (x1, y1) = (6, 10) and (x2, y2) = (8, 12).
Now, we will find out the slope of the function using the given formula:
Slope of the equation(m) = (y2 – y1) / (x2 – x1) = 12 – 10 / 8- 6 = 2 / 2 = 1.
Now, we put the values in the point slope form to obtain our required answer:
y – y1 = m (x – x1)
y – 10 = 1 (x – 6)
y = x + 4.
Thus, the required linear function is y = x + 4.
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