# Linear Function

In mathematics, a function is an expression, rule, or law that establishes a relationship between one independent variable and another dependent variable. Linear functions are algebraic equations whose terms are constants or the product of constants and single variables. When you draw a graph of a linear function it is a straight line__.__ For example, y = 5x + 7 is a linear function because it depicts a straight line on a coordinate plane.

## What is Linear Function?

A linear function is one whose graph is a straight line. This indicates that the function contains one or two variables with no exponents or powers. In order for a function to remain linear, all variables must be constants or known variables.

A function is defined as F(b), where b is an independent variable upon which the function depends. A linear function graph consists of a straight line whose expression or formula is as follows;

** y equals f(x) = ax + b**

There is one independent variable(x) and one dependent variable(y) in this equation. Examples of Linear Functions

- f(x) = 4x – 5
- f(x) = -7x – 0.45
- f(x) = 5

## Identifying Linear Functions

By understanding the following points thoroughly, you will be able to identify linear functions easily.

- The function must have one or two real variables, which is the first requirement. It must be a known variable or constant if another variable is present.
- Because only the C and r are real variables, and the pi is a constant, the function C = 2 * ℼ * r is a linear function.
- The second point is that no variable should have an exponent or a power. They can’t be squared, cubed, or transformed into any other shape. In the numerator, all variables must be.
- The function must also graph to a straight line as the third requirement. A curve of any kind does not qualify as a linear function.

## Calculating Linear Functions With Two Ordered Pairs

1. Using the formula **m=**, calculate the slope using the two ordered pairings(x1,y1) and (x2,y2).

- Substituting the slope and one of the ordered pairs into f(x)=mx+b and solving for b yields the y-intercept.
- In the function f(x)=mx+b, substitute the slope and y-intercept.

## Examples

**Example 1**: Find the slope of the line whose coordinates are (3,6) and (5,2).

**Solution:** We have,(x1, y1) = (3, 6) and (x2, y2) = (5, 2)

The slope of a line formula is **m=**

m=

m=

m= -2

**Examples 2**: Find an equation of the linear function given f(3) = 4 and f(5) = 8.

**Solution**: Now let us write the two ordered pairs

f(3) = 4 f(5) = 8

(x1, y1) =(3, 4) and (x2, y2) = (5, 8)

Find the slope for (3,4) (5,8)

slope=

=

= =2

∴ Slope = 2

In the equation, substitute the value of slope and y intercept , write an equation like this: y = mx+c

4 = (2) (3) + b

4 = 6 + b

b = 4- 6

b = -2, which is a y-intercept.

y = mx+b

y = (2) (x) – 2

In function Notation: f(x) =2x – 2

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