Linear Functions

You should know how to write an equation before you learn about linear functions. The only difference is the function notation. Knowing an ordered pair written in function notation in necessary too. f(a) is called a function. These graphs have a straight line.

y = f(a) = p + q(a)

It has one independent and one dependent variable. The independent variable is a and the independent one is y. P is the constant term or the y-intercept it is also the value of the dependent variable when a = 0, q is the coefficient of the independent variable as known as slope which gives the rate of change of the dependent variable. It can be calculated using a linear function calculator too.

Linear Function Graph

Linear function

A normal ordered pair A function notation ordered pair
(a,b) = (2,5) f(a) = y coordinate, a=2 and y = 5, f(2) = 5

Let’s move on to see how we can use function notation to graph 2 points on the grid.

  • Relation: It is a group of ordered pairs.
  • variable: A symbol that shows a quantity in a math expression.
  • linear function: If each term is either a constant or It is the product of a constant and also (the first power of) a single variable, then it is called as an algebraic equation.
  • function: A function is a relation between a
  • set of inputs and a set of permissible outputs.
  • It has a property that each input is related to exactly one output.
  • steepness: The rate at which a function deviates from a reference
  • direction: Increasing, decreasing, horizontal or vertical.

Examples of Linear Function:

Graphing of linear functions needs to learn linear equations in two variables.

Example 1 : Let’s draw a graph for the following function:

F(2) = -4 and f(5) = -3


  • Let’s rewrite it as ordered pairs(two of them).
  • f(2) =-4 and f(5) = -3

(2, -4) (5, -3)

How to evaluate the slope of a linear Function?

Let’s learn it with an example:

Example 2 : Find the slope of a graph for the following function.

f(3) = -1 f(-8) = -6


    • Let’s write it again as ordered pairs

f(3) =-1 and f(8) = -6

(3, -1) (8, -6)

    • we will use the slope formula to evaluate the slope

(3, -1) (8, -6)

(x1 , y1) (x2 , y2)

    • Slope Formula = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
\(\frac{-6-(-1)}{-8-(-3)} =\frac{-5}{-11}\)

= 5/11 is the slope for this function.

Linear Equation Formula:

  • y = mx+c

Linear Function-1

Example 3: How to write the equation of a linear Function.

Find an equation of the linear function given f(2) = 5 and f(6) = 3.

Solution: Let’s write it in an ordered pairs

    • f(2) = 5 f(6) = 3

(2, 5) (6, 3)

    • Find the slope.

(2, 5) (6, 3)

\(\frac{y_{2}-y_{1}}{x_{2} – x_{1}} = \frac{3-5}{6-2} = \frac{-2}{4} =\frac{-1}{2}\)

Slope = -1/2

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

5 = -(½) (2) + b

5 = -1 + b

b = 5 + 1

b = 6, which is a y-intercept.

    • y = mx+b

y = -(½) (x) + 6

  • In function Notation: f(x) = -(½) (x) + 6

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