Linear Functions

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. 

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.