What are Linear Functions?

The word linear means line. The linear functions are straight lines in the graph, and they can be written as equations. For example, if $f\left(x\right)=ax+b$ is a linear function, then $y=ax+b$ is the corresponding linear equation. An algebraic expression with variables having degree 1 is said to be linear. By calling it a function it means that it is a relation between a collection of inputs in a set and a collection of outputs in a set and it should be in such a way that each input has exactly one output.

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x², x^1/2 or y² it is not linear. The exponent over the variables should always be 1.

Comparison between Linear Functions and Quadratic Functions

The equation x+3=0 is linear whereas the equation x²+2=0 is not linear but quadratic because the degree is 2.

Linear functions have one variable which is independent(x) and another variable which is dependent(y).

How Linear Functions are used in Real Life?

It is helpful in finding the slope of the Leaning Tower of Pisa, the steepness of the mountain, and so on.

Application Problem with Linear Function

If the cost of 1 chocolate is 2 bucks, then the cost of 2 chocolates is $2\times 2=4$ bucks. What is the cost of 5 chocolates? Let x denote the chocolates and y denote the cost. The cost is dependent on the number of chocolates bought. So, y is dependent on x. It can be written as y=2x because the cost is twice the number of chocolates. So, 5 chocolates would cost y=$2\times 5$ which is 10 bucks.

How to represent Linear Functions in the Graph?

As x and y are variables, it keeps changing for different values given. Here x is input, and y is output. If x=0 then y becomes 0. For x=1, y=2. For x=-2 it is y=-6. For every input, there is an output.

Write this as an ordered pair (x,y). For x=0, the function of x at 0 is 0 and it is written as (0,0). Similarly, the ordered pairs are (-2,-4), (-1,-2), (1,2), (2,4), (3,6), and so on. Graphing linear functions means to plot the points on the graph and join them by using a straight line.

Let us try adding a constant 2 to y=2x. This gives y=2x+2. Try giving values to x and obtain y. If x=0, the ordered pair is (0,2). This is because y takes the value 2 when x is 0. Similarly, for x= -1, the ordered pair is (-1,0). For x= 1, the ordered pair is (1,4). Let us plot these coordinates graphing linear functions and then join the points. Here also the straight line is obtained.

Here, the graph cuts the y axis at 2. So, this is called the y-intercept.  It is also calculated by setting x=0. Similarly, x-intercept can be found by setting y=0. What is x-intercept here? It is -1.

$\begin{array}{l}0=2x+2\\ -2=2x\\ \frac{-2}{2}=x\\ -1=x\end{array}$

In y=2x+2, the coefficient of x is 2 and is called the slope. It determines how steep the line is and also talks about the direction of the line. It is denoted by the letter m. The slope-intercept form of a line is y= mx+b. Compare it with the linear equation y=3x+9 and find the slope, the x-intercept, and the y-intercept.

Slope

A slope is a number, and it can be positive, negative, zero, or undefined. If it is positive, then the line goes up from left to right.

If it is negative, the line goes down from left to right.

When the slope is zero, the line is horizontal. When the slope is undefined, the line is vertical. But this is not a linear function because x takes infinite values and by the property of functions, x can have only one output.

Calculating Slope

For finding slope m, two points are needed. Suppose the two points are (-1,-1) and (1,1). Here x=y. In the 1^st point, the x-coordinate is -1 and in the 2^nd point, the x-coordinate is 1. In the 1^st point, the y-coordinate is -1 and in the 2^nd point, the y-coordinate is 1. To find m,

• Subtract x coordinate of 2^nd point from x coordinate of 1^st point. It is 1-(-1) =2 here.
• Subtract y coordinate of 2^nd point from y coordinate of 1^st point. It is 1-(-1) =2 here.

The slope is 1 here.

How to write a Linear Equation in Slope-Intercept Form?

Take a linear equation 3x+4y= -5. The y term should be before the equal sign.

Step 1: Subtract 3x on both sides.

3x+4y-3x=-5-3x

This gives, 4y=-5-3x

Step 2: Divide 4 on both sides.

$\begin{array}{c}\frac{4y}{4}=\frac{-5-3x}{4}\\ y=\frac{-3x}{4}-\frac{5}{4}\end{array}$

Now the required form is y= mx+b. Here the slope m=$\frac{-3}{4}$ and b=$\frac{-5}{4}$.

Finding Zeroes of Linear Functions

It involves 2 steps:

Step 1: Substitute y= 0.

Step 2: Solve for x.

Find the zeroes of the linear function y=4x-16 by following the above steps.

Solution: By considering y= 0 the solution is, 0=4x-16.

To solve for x,

1. Add -4x on both sides. This gives -4x= 4x-16-4x.

This becomes -4x= -16.

2. Divide both sides by -4.

This gives,

$\begin{array}{c}\frac{-4x}{-4}=\frac{-16}{-4}\\ x=4\end{array}$

The zero is (4,0).

How to write the Equation of a Line when the Slope and a Point are Given?

For writing this, there is a point-slope form. The point-slope form is given as $y-y_1=m\left(x-x_1\right)$.

Find the equation of the line using the slope m= 3 and the point (3,4).

Step 1: Subtract the x coordinate (here it is 3) of the point (3,4) from x and y coordinate (here it is 4) of the point (3,4) from y in y=mx.

This gives y-4=3(x-3).

Step 2: Solve for y.

For this, add 4 on both sides.

$\begin{array}{c}y-4+4=3\left(x-3\right)+4\\ y=3x-9+4\\ y=3x-5\end{array}$

Thus, the equation of a line using one point and slope can be found.

Formulas

The formula for the slope of a linear equation is $m=\frac{y_2-y_1}{x_2-x_1}$, where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are two points on the line.

The slope-intercept form of the equation of a line is: $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept of the line.

The point-slope form of the equation of a line is: $y-y_1=m\left(x-x_1\right)$, where $m$ is the slope and $\left(x_1,y_1\right)$ is a point on the line.

Common Mistakes

The equations which have degree 1 alone are linear equations. Higher powers are not linear.

Context and Application

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for   

• Bachelor of Science Mathematics       
• Master of Science Mathematics