Linear Functions: Definitions, Formulas, Problems and Discussion, Along with Other Matters

Linear Functions – The existence of linear functions is not only related to the world of mathematics , you know , but can also be applied in the world of economics, or more precisely economic mathematics. Therefore, even though it is difficult to calculate, the result or impact is very large. However, in this article, we will discuss linear functions in the world of mathematics .

Linear functions in the world of mathematics will always be closely related to variables, coefficients and constants. You must still remember this material when he was in grade 10? Yep, these three things will later affect the process of presenting functions in graphical form. Then, what is a linear function? What about the formula and its application in the problem? What other things are related to this linear function? So, so that You understands these things, let’s look at the following review!

What is a Linear Function?

Basically, the definition of a linear function is a relation that pairs every member in set A with another member in set B. All members in set A must be paired with members in set B. This linear function is also called the Line Equation Straight (PGL). As previously stated, functions in the world of mathematics are closely related to their constituent elements, namely in the form of variables, coefficients, and constants.

So, a linear function is a function that forms a graph in a straight line. This linear function is also a function that has the highest rank with a variable equal to one.

Linear Function Formulas

The general form of a linear function is as follows,

f : x → mx + c or it can also be

f(x) = mx + c or it can also be

y = mx + c

Information:

m = gradient or slope

c = constant

Well, this linear function will not be far from what is called graphical depiction, so when you have to do it, you have to pay attention to the following steps.

  • Determine the point of intersection with the x axis, then y = 0, obtained from coordinates A (x1, 0)
  • Determine the point of intersection with the y axis, then x = 0, obtained from coordinate B (0, y1)
  • Connecting two points A and B, so that it will form a straight line linear equation which is then written as y = ax + b .
  • If b is positive , then the linear function will be drawn a line from the bottom left to the top right .
  • If b is negative , then the linear function will be drawn a line from the top left to the bottom right .
  • If b is 0, then the linear function will be drawn a line parallel to the X axis.
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Example Questions and Discussion

Problem 1

There is a linear function that is f(x) = 6x + b. Determine the form of the function if it is known that f(4) = 8.

Completion:

f(x) = 6x + b

f(4) = 6 x 4 + b = 8

8 = 6 x 4 + b

b = 8 – 24

b = -16

f(x) = 4x – 16

Problem 2

There is a linear function that is f(x) = 10x + b. Determine the form of the function if it is known that f(5) = 15.

Completion:

f(x) = 10x + b

f(5) = 10 x 5 + b = 15

15 = 10 x 5 + b

b = 15 – 50

b = -35

f(x) = 10x – 35

Understanding Other Things in Linear Functions

In the world of mathematics, the existence of this “function” can be interpreted as ‘the relationship between one variable and another, where each of these variables influences one another’. This relationship also states that each domain member (first member/region) has a relationship with one and only one range member (second member/range). In this function there are several components, namely in the form of:

  • Variable, namely a quantity whose value in a problem can change. Variables can be divided into two, namely the independent variable and the dependent variable. Independent variables are variables that explain other variables; while the dependent variable is the variable explained by the independent variable.
  • Constants, namely numbers or numbers that form a function but are not related to a variable.
  • Coefficients, namely numbers or numbers that participate in forming a function and are related to a variable in the function concerned.

Now, back to the discussion of linear functions, its existence is a function that has the highest rank of the independent variable which is one (1). The general form is

y = ax + b

Information:

y = dependent variable

x = independent variable

a = coefficient

b = constant

In this general form, namely y = ax + b, states that y is a function of x. That is, the size of the x value will affect the size of the y value.

Representing Functions With Graphs

In presenting the function with this graph, it can be done in two ways, namely the list method and the mathematical method. Here’s the description.

How to Register

For example, there is a problem in the form of “Draw a graph of an equation y = 2x + 10”. Then for the implementation of the list method is:

The Mathematical Way

For example, there is a problem in the form of “Draw a graph of an equation y = 2x + 10”. Then for the implementation of the list method is:

  • The point of intersection with the y axis if x = 0, then y = 10, so the point of intersection with the y axis will occur at (0.10)
  • The point of intersection with the x axis if y = 0, then 0 = 2x + 10, x = -15, so the point of intersection with the y axis will occur at (-5,0)
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So if it is drawn into a graph, it is as follows:

Slope of a Line

Since the linear function is closely related to the graph, the line must be sloped properly. In this case, if it is known that there are two points with coordinates (x1,y1) and (x2,y2), which are located in a straight line, then the slope of the line becomes:

There are several things that must be considered, namely:

  • a > 0; the line of the linear equation will move from the bottom left to the top right.
  • a < 0; the line of the linear equation will move from the top left to the bottom right.
  • a = 0; the line of the linear equation will move from the magnitude of the constant b so that it is parallel to the x-axis to the left or right.
  • b > 0; the graph of the linear equation will intersect the y-axis which has a positive value.
  • b < 0; the graph of the linear equation will intersect the y-axis which has a negative value.
  • b = 0; the graph has no point of intersection with the y axis, so the graph will move from the starting point or point 0.

Relations Between Straight Lines

In a linear function that “requires” that there is a straight line between the two sets, it has the following relationships:

  • Two Lines Intersect

Two straight lines will coincide with each other, if there is an equation of one line which is the equation of the other line.

  • Two Parallel Lines

Two straight lines will be parallel to each other if their slopes (gradients) are the same.

  • Two Intersecting Lines

Two lines will intersect each other, if the slopes of the two lines are different or do not have the same magnitude.

  • Two Intersecting Lines Perpendicular

Two lines will intersect at right angles to each other if their slopes are opposite to each other with opposite signs.

So, that’s a review of what a linear function is and its formula. Does You understand how to solve problems with linear functions?

Source:

Istiningrum, Andian Ari. Linear Functions .