Linear function

Author: Leandro Alegsa

13-11-2022

In basic mathematics, a linear function is a function whose graph is a straight line in 2-dimensions (2D, see images). An example is: y=2x–1. In higher mathematics, a linear function often refers to a linear mapping.

Graph

The graph of a linear function is a straight line. In Cartesian coordinates following applies (x|y)

$y=m\cdot x+n$

with real numbers and where (the abscissa) is an independent variable and (the ordinate) is the dependent variable.

There are numerous other naming conventions for the function term, such as ax+b, mx+c, mx+b or mx+t. In Austria, y=kx+d is often used, but in Switzerland, y=mx+q. In Belgium, one also finds y=mx+p or y=kx+t.

This representation is also called the normal form of a linear function. Its two parameters can be interpreted as follows:

The number indicates the slope of the line.
The number is the y-axis or ordinate intercept, inhomogeneity, or displacement constant.

The graph of a linear function is never parallel to the y-axis, because this would mean that one would have more than one associated with it, which would contradict the definitionally required (right-)uniqueness of a function.

Determination of the function term from two points

It is assumed that the points $(x_{1}|y_{1})$ and lie $(x_{2}|y_{2})$ on the graph of the linear function and are distinct from each other.

The slope can be calculated with

$m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.$

The y-axis intercept is given by

$n=y_{1}-m\cdot x_{1}$ or $n=y_{2}-m\cdot x_{2}.$

The function term f(x) is given by

$f(x)={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot x+\left(y_{1}-{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot x_{1}\right)$

or more simply by

$f(x)={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot (x-x_{1})+y_{1}.$

Slope of a linear function through two given points

Questions and Answers

Q: What is a linear function in basic mathematics?

A: A linear function in basic mathematics is a function whose graph is a straight line in 2-dimensions (2D).

Q: Can you provide an example of a linear function?

A: An example of a linear function is y=2x-1.

Q: What does a linear function refer to in higher mathematics?

A: In higher mathematics, a linear function often refers to a linear mapping.

Q: Is a linear function always represented by a straight line?

A: Yes, a linear function is always represented by a straight line.

Q: Can a linear function have multiple solutions?

A: No, a linear function can have only one solution because it is a straight line.

Q: Is y=3x+2 a linear function?

A: Yes, y=3x+2 is a linear function because its graph is a straight line.

Q: In how many dimensions is the graph of a linear function represented?

A: The graph of a linear function is represented in 2 dimensions (2D).

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