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.
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=
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
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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