Important Notes on Linear Functions

In Mathematics, the function is an expression that shows the relationship between one independent variable and another dependent variable. Alternatively, in other words, a function is a relation between a set of inputs and the permissible outputs. It is commonly denoted by letters f, g, and h as f(x), g(x), and h(x), respectively, where x is the input variable. There are many different kinds of functions in mathematics, including injective function, surjective function, Polynomial function, algebraic function, Identical function, linear function, quadratic function, Periodic function, etc. In this article, let us focus on linear functions, how they are represented and calculated, and how to graph them. 

What is meant by Linear Functions?

A linear function is a type of function which represents a straight line in a coordinate plane. The form of a linear function is f (x) = mx + c, where m is the slope of the line, c is the y-intercept, x is the independent variable, and y is the dependent variable. These variables are real numbers and do not have any exponent or power. The linear function is also algebraic. All the linear functions are represented by equations, presented by slope and y-intercept. No matter what direction the line goes on plotting, it should be a straight line if the function is linear.                                  

Real-life analysis of Linear Functions

We all know that mathematics has its application to the entire universe. Here are a few situations where linear function has its application in our day-to-day life. 

For example, Suppose you want to bake your favourite cake. You add ½ cup of butter, 2 cups of flour, three eggs, sugar, and milk. However, this time your best friend is also there. So, it would be best if you bake a cake twice as big as you usually bake. So you will have to double the number of ingredients. 

You are talking on the telephone for 1 hour. The longer you talk, the higher the telephone charge. Here there is a relationship between the time and the telephone charge. 

Imagine you are going shopping. There is a direct relationship between buying and the money you spend.

Calculation of Linear Functions

We generally use the slope-intercept form to calculate the Linear Function. 

Let, ( x1, y1 ) and ( x2, y2 ) be any two points. On calculating the slope

    m = ( y2 – y1 ) / ( x2 – x1 )

With this, the equation of a linear function can be calculated using slope-intercept form

y – y1= m( x – x1)

Look at the example given below;

Example:Calculate the linear function of a line with points (4, 20) and (6, 32)

Solution: x1 = 4 , x2 = 6 , y1 = 20 , y2 = 32

On calculating slope,    m = (y2 – y1)/ (x2 – x1) 

   m = (32 – 20) / (6 – 4)

   m = 12/2 = 6

Using slope intercept form y – y1 = m( x – x1)

    y – 20 = 6 ( x – 4)

    y – 20 = 6x -24

    y = 6x – 4 

So the linear function of the line is y = 6x – 4.

Graphing a Linear Function

The linear function is represented as a straight line in a graph. The graph of a linear function f(x) = mx + b is an increasing line if m is greater than 0, decreasing line if m is less than zero, and horizontal line if m is zero. If we know any two points, we can plot them and join them. By extending it, we get a straight line. Generally, there are two methods to graph a linear function.

• Finding and using two points

• Using slope and y-intercept

Graphing linear function using two points

For finding any two points on a linear function f(x) = mx + c, we have to assume some random values for x and substitute them in the equation to get the respective value of y. 

For example; The linear function is f(x) = 2x + 3

Take any random value for x. On assuming x equal to -1 and 4 we get f(x) = 1 or f(x) = 7 respectively. Note down the corresponding values of y we got when substituting random values of x.  

So, the two points are ( -1, 1 ) and  ( 4, 7).

Plot these points on the graph, join them and extend them on both sides. 

Graphing a linear function using

First, make sure that the equation is in slope-intercept form. Graph the y-intercept c on the y axis. Next, using slope, find the second point corresponding to that intercept. The slope is the coefficient of x, which is m, and y is c.

Plot the y-intercept ( 0, c) in the xy axis. From the y-intercept, find the next point using the slope. The slope will give the direction on the rise and run of the units. Connect the two points found using the y-intercept. 

Conclusion

A function represents the relationship between two variables. There are two ways to find the linear function. One is through any two points given, and the other is through the slope-intercept form. The standard equation of a linear function is y = mx + c where y is a dependent variable, x is also a variable and can be manipulated according to the values of y, m is the coefficient of x, which is also the slope, and c is the constant which is also the y-intercept. 

 Linear functions are straight lines when graphed on a coordinate plane. For calculating the linear function, there are two methods. It uses any two points in the line and uses the slope-intercept form. No matter what direction the line goes, be it upward, downward, left, or right, it should be straight to be a linear function.