Linear Functions

Dhristi JEE 2022-24

Graphical representation of data is not just an important aspect of mathematics but a key element in the field of economics as well. The linear function is one such topic that will be present in both subjects. This entire topic revolves around a single equation that represents the increase and decrease of a variable when the other variables are kept constant. 

Linear Function Definition? 

It is a polynomial function with a maximum degree of 0 to 1. It is represented on the graph with a straight line that usually has a slope.  To better understand this function, let’s first look at its equation. 

y = f(x) = a + bx

Here ‘y’ represents the Y-axis in the graph and is a dependent variable as its value depends on the value of ‘x’. In contrast, ‘x’ here is an independent variable and represents the X-axis. In this equation, ‘a’ is a constant and when ‘x=0’ it is also the value of ‘y’. Thus ‘a’ is also called the y-intercept in this equation. The other constant in this equation is b’’, and its value is never 0. 

Do not confuse between a linear and a nonlinear function as they are very different from each other. The most evident difference between a linear and a nonlinear function is when their values are put on a graph the former forms a straight line while the latter forms a curve. 

For example, a graphical representation of linear function looks like.

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Whereas, a nonlinear function graph looks something like. 

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Different Forms of this Equation

y = f(x) = a + bx

While the above-mentioned formula is the general way of representing linear function, it can be written in several different ways as well. Few examples are

y - 4 = 2(x + 1)

y = 2x - 5

\[\frac{y}{3}\] = 2

2x = 5

y + 3x - 2 = 0

All of these are some different ways of writing the linear function formula. While if the equations consists of even a single variable with an exponent or square roots and cube roots, which is not a linear but a nonlinear function.  Some of its examples are 

y\[^{2}\] + 3 = 0

x\[^{2}\] + 2 = y

Formulation of a Linear Function through Table

The table below shows both normal and function form of the ordered pairs. 

Normal ordered 

Function notation ordered 

(a,b) = (3,5)

f (a) = y coordinate, a=3 and y=5. i.e. f (3)=5

By utilising the table, one can verify it by checking the values assigned to x and y. In this type of function the rate at which y changes in respect to x remains constant. That means 

x - y = c (Here c will be the constant)

So by putting the values of x and y, we get,

5 – 3 = 2 

This rate of change is called slop, and slope for this example will be 2. 

Now let’s consider the following table,













From here, it can be observed that the rate of change between x and y is indeed 2. Now, this can be expressed by a linear function, that is –

y = x + 2

Linear Function Examples 

For a graphical representation of this function, one needs to learn linear equations with two variables. Only by solving a linear equation one can find the different values of x and y and put them in a linear function graph. 

Example Number 1

Frame an equation from the given function that is f (2) = 2 and f (4) = -4

First, find the slope by using the formula

\[\frac{y-y_{1}}{x-x_{1}}\] = b

That is,

b= (-4-2)/(4-2) =-6/2 = -3

b= -3

Now put any pair of (x, y) and the value of b in 

y = f(x) = a + bx

2 = f(2) = a + (-3 x 2) 

Lastly, solve this equation for a.

2 = a – 6

a = 2+6

a = 8, the y-intercept

So, its equation will be

y = 8 – 3x

Example Number 2

Draw a graph for the following function

f (0) =7 and f (4) = 4

First rewrite them as order pairs, that is 

(0, 7) and (4, 4)

Now mark those points on the graph and connect them with a line, the way it is represented below. 

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To understand the concept of a linear function in detail, check Vedantu’s online classes and live tuitions. Avail their sample papers and practice these kinds of sums daily. It doesn’t matter if your concept about this topic is crystal clear if you do not practice these sums daily. Take their mock tests after practising for a few days to understand how much you have improved and where you need more work. 

FAQs on Linear Functions

1. What is a Linear Function?

Ans. When two or more different notions have a distinct relation between them, they are referred to as linear function. This mathematical term can be found in calculus as well as other related areas where it is represented with a straight line. Its formula shows the relationship between x-axis, y-axis and its slope. Its slope can be positive, negative, undefined or zero. 

2. What Kind of Graphs Does Nonlinear Function Forms?

Ans. As its name suggests, its graphical representations are never straight lines; thus, they cannot tell the relationship between two variables in economics. A nonlinear function has variables with exponents and square roots or cube roots. This addition drastically changes the relationship between x and y. There is also no distinct slope, and unlike its linear counterpart, its equations form curves, circles, hyperbolas and parabolas. These equations form curved lines that are constantly changing their direction. 

3. What is the Slope in a Linear Function?

Ans. The slope in this type of function is constant. In mathematics, the steepness and direction both are described by the slope of a line. Its line’s direction can be increasing or decreasing, vertical or horizontal depending upon its value.

In its equation, that is 

y = f(x) = a + bx

Variable ‘b’ represents the slope. 

We say a line is increasing when it goes up from left to right implying that its slope is positive or b ˃ 0. In contrast, if the goes down from left to right, it is decreasing or b ˂ 0. If the line is vertical, its slope is said to be undefined, and when it is horizontal, its slope is zero.