Linear function (calculus)

In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph is a line in the plane.^[1]

Properties of Linear Functions

A linear function is a polynomial function of first degree with one independent variable х, i.e.
$y(x)=ax+b$ (In the USA, the coefficient letter a is usually replaced by m.)
- To use the function or graph the line, the coefficient letters a and b must be given as actual real numbers. For example: $y(x)=2x-1$ . Here a=2 and b=-1.
- In the function, x is the independent variable and y is the dependent variable.
- The ___domain or set of allowed values for x of a linear function is $\mathbb {R}$ (all real numbers). This means that any real number can be substituted for x.
- The set of all points: (x,y(x)) is the line.
- Since two points determine a line, it is enough to substitute two different values for x in the linear function and determine

y for each of these values (see videos below).

Graph of the linear function: y(x)=-x+2

Because the graph of a linear function is a line, a linear function has exactly one intersection point with the у-axis. This point is (0,b).
A nonconstant linear function has exactly one intersection point with the х-axis. This point is (-b⁄a,0).
From this, it follows that a nonconstant linear function has exactly one zero or root. That is, there is exactly one solution to the equation a'x+b=0. The zero is x=-b⁄a.

There are three standard forms for linear functions.

General Form

$Ax+By=C$ where $A\neq 0$ and $B\neq 0$ .

Properties of the general form

The general form has 2 variables x and у and 3 coefficient letters A, B, and C.
To use the function or graph the line, the coefficient letters A, B and C must be given as actual real numbers: 3x-2y=1. Here A=2, B=-2 and C=1.
This form is not unique. If one multiplies A, B and C by a factor k, the coefficients change, but the line remains the same.

- Example: For k=3, it follows that (e.g.) 3x-2y=1 and 9x-6y=3 are the same line.
- Example: For k=-1⁄π, it follows that -3πx+2πy+π=0 and 3x-2y=1 are the same line.

This general form is used mainly in geometry and in systems of two linear equations in two unknowns.

Example: 3x-2y=1 and 6x-4y=2 are the same linear function, i.e. their graph is the same line.
- In the first equation the coefficients are: A=3, B=-2 and C=1.
- In the second equation the coefficients are: A=6, B=-4 and C=2.
- Notice that the second coefficients are all twice the first equations. This means the factor is k=2.
- Further, solving both of these equations for y gives the same slope-intercept form of this line.
y=1,5x-0,5

Slope-Intercept Form

$y(x)=ax+b$ or $y=ax+b$ where ${a\neq 0}$ .

Properties of the Slope-Intercept Form

The slope-intercept form is also called the explicit form because it defines y(x) explicitly (directly) in terms of x.
The slope-intercept form has 2 variables x and у and 2 coefficient letters а and b.
To use the function or graph the line, the coefficient letters must be written as actual real numbers. For example: y(х)=-2$
The slope-intercept form is unique. That is, if the value of either or both of the coefficient letters a and b are changes, th$
Every linear function can be written uniquely in slope-intercept form.
Intercepts (intersections of the line with the axes)
The constant b is the so-called у-intercept. It is the y-value at which the line intersects the y-axis. This is$
The number -b⁄а is the [root] or [[http://en.wikipedia.org/wiki/Zero_o$
0=a•x+b
The number -b⁄а is the [root] or [[http://en.wikipedia.org/wiki/Zero_o$
0=a•x+b

a•x+b=0

a•x = -b

x=-b⁄a

This means that the point (-b⁄a,0) is both a point on the line and a point on the x-axis. So it is the point where the line intersects the x-axis.

The coefficient а is the so-called slope of the line and is a measure of the rate of change of the linear function. Since a is a number (not a variable), this rate of change is constant. Moving from any point on the line to the right by 1 (that is, increasing x by 1), the y-value of the function changes by a. See slope.

Example: $y(x)=-2x+4$ where a = -2 and b = 4

(0,b) = (0,4) is the intersection of the line and the у-axis
-b⁄а,0) = (-4⁄-2,0) = (2,0) is the intersection of the line and the х-axis and
а = -2 is the slope of the line. For every step to the right (х increases by 1), the value of у changes by -2 (goes down).

Vector-Parametric Form

Parametric form:

x(t) = {b_1}+{a_1}t

y(t) = {b_2}+{a_2}t

where $a_{1}\neq 0$ .

Properties of the vector-parametric form

Vector-parametric form has 1 parameter t, 2 variables x and у, and 4 coefficients а₁, а₂, b₁, and b₂.
The coefficients are not unique, but they are related.
The line passes through the points (b₁,b₂) and (b₁+a₁,b₂+a₂).
The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with t=time).
Engineers tend to use parametric notation and the letter t for the parameter; mathematicians use vector notation and the letter λ.
This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.
This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.

Example: ${X}=({-1},{1})+t({2},{3})$

Here: a₁ = 2 and a₂ = 3 and b₁ = -1 and b₂ = 1
The line passes through the points> (b₁, b₂)=(-1,1) and (b₁+a₁,b₂+a<s$
The parametric form of this line is:
x(t) = {-1}+{2}t

y(t) = {1}+{3}t
The slope-intercept form of this line is: y(x)=1,5x+2,5 (solve the first parametric equation for t and substitute in$
One general form of this line is: -3x+2y=5.

References

James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9

External links

^ Stewart 2013, p. 23

[1] Stewart 2013, p. 23

[1]