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

Graph of the linear function:

y (x) = - x + 2

A linear function is a polynomial function with one independent variable $x$ of degree at most one, i.e. $f(x)=ax+b$ .^[2] Here $x$ is the independent variable. The graph of a linear function, that is, the set of all points whose coordinates have the form ( $x$ , $f(x)$ ), is a line, which is why this type of function is called linear. Some authors, for various reasons, also require that the coefficient of the variable (the $a$ in $ax + b$ ) should not be zero.^[3] This requirement can also be expressed by saying that the degree of the polynomial in this polynomial function is exactly one, or by saying that the line which is the graph of a linear function is a slanted line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, $f(x)=b$ , will be considered to be linear functions (their graphs are horizontal lines).

The ___domain or set of allowed values for $x$ of a linear function is the entire set of real numbers $R$ . This means that any real number can be substituted for $x$ .

Because two different points determine a line, it is enough to substitute two different values for $x$ in the linear function and determine $f(x)$ for each of these values. This will give the coordinates of two different points that lie on the line. Because $f$ is a function, this line will not be vertical.

Since the graph of a linear function is a nonvertical line, this line has exactly one intersection point with the $y$ -axis. This point is $(0, b)$ .

A the graph of a nonconstant linear function has exactly one intersection point with the $x$ -axis. This point is $(.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠−b/a⁠, 0)$ . 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 $ax + b = 0$ . The zero is $x =$ ⁠−b/a⁠.

The points on a line have coordinates which can also be thought of as the solutions of linear equations in two variables (the equation of the line). These solution sets define functions which are linear functions. This connection between linear equations and linear functions provides the most common way to produce linear functions.

There are many standard forms of linear equations, but only three will be examined to see how to obtain linear functions:

General form
Slope-intercept form
Parametric form

Slope-intercept form

The slope-intercept form of a linear function is an equation of the form

y(x)=ax+b

.

The slope-intercept form has two variables $x$ and $y$ and two coefficients $a$ and $b$ . 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 of a linear function is unique. That is, if the value of either or both of the coefficient letters $a$ and $b$ are changed, a different function is obtained.

The constant $b$ is the so-called $y$ -intercept. It is the $y$ -value at which the line intersects the $y$ -axis.

The coefficient $a$ is the slope of the line, which measures of the rate of change of the linear function. Since $a$ is a constant, this rate of change is constant. Moving from any point on the line to the right by one unit (that is, increasing $x$ by 1), the $y$ -value of the function changes by $a$ .

For example, the slope-intercept form $y(x)=-2x+4$ has $a = -2$ and $b = 4$ . The point $(0, b) = (0, 4)$ is the intersection of the line and the $y$ -axis, the point $(⁠ - b / a ⁠, 0)$ = (⁠−4/−2⁠, 0) = (2, 0)}} is the intersection of the line and the $x$ -axis, and $a = -2$ is the slope of the line. For every step to the right ( $x$ increases by 1), the value of $y$ changes by −2 (goes down).

General form

The general form for the equation of a line is:

Ax+By=C.

When $B\neq 0$ this equation may be solved for the variable $y$ and thus used to define a linear function (namely, $y=-{\tfrac {A}{B}}x+{\tfrac {C}{B}}=f(x)$ ). While all lines have equations in the general form, only the non-vertical lines have equations which can give rise to linear functions.

The general form has 2 variables $x$ and $y$ and 3 coefficients $A$ , $B$ , and $C$ .

This form is not unique. If one multiplies $A$ , $B$ and $C$ by a constant factor $k$ , the coefficients change, but the line remains the same. The linear function obtained from this form is unique since it depends only on the coordinates of the points on the line. For example, $3 x - 2 y = 1$ and $9 x - 6 y = 3$ are general forms of the equation of the same line which is associated with the linear function $f(x)={\tfrac {3}{2}}x-{\tfrac {1}{2}}$ .

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

Parametric form

The parametric form of a line consists of two equations:

x(t)={b_{1}}+{a_{1}}t

y(t)={b_{2}}+{a_{2}}t

where $a_{1}\neq 0$ .

The parametric form has one parameter $t$ , two variables $x$ and $y$ , and four coefficients $a 1$ , $a 2$ , $b 1$ , and $b 2$ . The coefficients are not unique, but they are related.

The line passes through the points $(b 1, b 2)$ and $(b 1 + a 1, b 2 + a 2)$ .

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.

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

Here: $a 1 = 2$ and $a 2 = 3$ and $b 1 = -1$ and $b 2 = 1$
The line passes through the points> $(b 1, b 2) = (-1, 1)$ and $(b 1 + a 1, b 2 + a 2)$
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: $y (x) = 1.5 x + 2.5$ (solve the first parametric equation for $t$ and substitute in$
One general form of this line is: $-3 x + 2 y = 5$ .

Notes

^ Stewart 2012, p. 23
^ Stewart 2012, p. 24
^ Swokowski 1983, pg. 34 is but one of many well known references that could be cited.

References

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

External links

[1] Stewart 2012, p. 23

[2] Stewart 2012, p. 24

[3] Swokowski 1983, pg. 34 is but one of many well known references that could be cited.

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