Linear function (calculus)

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

In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates with uniform scales) is a line in the plane.^[1] Their characteristic property that when the value of the input variable is changed, the change in the output is a constant multiple of the change in the input variable.

Linear functions are related to linear equations.

Properties

A linear function is a polynomial function in which the variable $x$ has degree at most one, which means it is of the form^[2]

f (x) = ax + b

.

Here $x$ is the 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 on the Cartesian plane (if over real numbers). That 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 defining the 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 $R$ , or whatever field that is in use. 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. If the value of either or both of the coefficient letters $a$ and $b$ are changed, a different line is obtained.

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 $(0, b)$ .

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 (also called a root). That is, there is exactly one solution to the equation $ax + b = 0$ ; it is $x =$ ⁠−b/a⁠.

Slope

The slope of a nonvertical line is a number that measures how steeply the line is slanted. The first derivative of a linear function, in the sense of calculus, is exactly this slope of the graph of the function. For $f (x) = ax + b$ , this slope and derivative is given by the constant $a$ . Linear functions can be characterized as the only real-valued functions that are defined on the entire real line and have a constant derivative.

In calculus, the derivative of a general function measures its rate of change. Because a linear function $f (x) = ax + b$ has a constant rate of change $a$ , it has the property that whenever the input $x$ is increased by one unit, the output changes by $a$ units. If $a$ is positive, this will cause the value of the function to increase, while if $a$ is negative it will cause the value to decrease. More generally, if the input increases by some other amount, $c$ , the output changes by $ca$ .

The fundamental idea of differential calculus is that any smooth function $f(x)$ can be closely approximated by a unique linear function near a given point $x=c$ . The derivative $f'(a)$ is the slope of this linear function, and the approximation is: $f(x)\approx f'(c)(x{-}c)+f(c)$ for $x\approx c$ .

Slope-intercept, point-slope, and two-point forms

A given linear function $\ell (x)$ can be written in several standard formulas displaying its various properties. The simplest is the slope-intercept form:

\ell (x)=mx+b

,

from which one can immediately see the slope m and the initial value $\ell (0)=b$ , which is the y-intercept of the graph $y=\ell (x)$ .

Given a slope m and a known value $\ell (a)=b$ , one writes the point-slope form:

\ell (x)=m(x{-}a)+b

.

In graphical terms, this gives the line $y=\ell (x)$ with slope m passing through the point $(a,b)$ .

The two-point form starts with two known values $\ell (a_{1})=b_{1}$ and $\ell (a_{2})=b_{2}$ . One computes the slope $m={\tfrac {b_{2}-b_{1}}{a_{2}-a_{1}}}$ and inserts this into the point-slope form:

\ell (x)={\tfrac {b_{2}-b_{1}}{a_{2}-a_{1}}}(x{-}a_{1}\!)+b_{1}

.

Its graph $y=\ell (x)$ is the unique line passing through the points $(a_{1},b_{1}\!),(a_{2},b_{2}\!)$ . This may also be written to emphasize the constant slope:

{\frac {y-b_{1}}{x-a_{1}}}={\frac {b_{2}-b_{1}}{a_{2}-a_{1}}}

.

Relationship with linear equations

Linear functions commonly arise from practical problems involving variables $x,y$ with a linear relationship, that is, obeying a linear equation $Ax+By=C$ . If $B\neq 0$ , one can solve this equation for y, obtaining

y=-{\tfrac {A}{B}}x+{\tfrac {C}{B}}=ax+b,

where we denote $a=-{\tfrac {A}{B}}$ and $b={\tfrac {C}{B}}$ . That is, one may consider y as a dependent variable (output) obtained from the independent variable (input) x via a linear function: $y=f(x)=ax+b$ . In the xy-coordinate plane, the possible values of $(x,y)$ form a line, the graph of the function $f(x)$ . If $B=0$ in the original equation, the resulting line $x={\tfrac {C}{A}}$ is vertical, and cannot be written as $y=f(x)$ .

The features of the graph $y=f(x)=ax+b$ can be interpreted in terms of the variables x and y. The y-intercept is the intial value $y=f(0)=b$ at $x=0$ . The slope a measures the rate of change of the output y per unit change in the input x. In the graph, moving one unit to the right (increasing x by 1) moves the y-value up by a: that is, $f(x{+}1)=f(x)+a$ . Negative slope a indicates a decrease in y for each increase in x.

For example, the linear function $y=-2x+4$ has slope $a=-2$ , y-intercept point $(0,b)=(0,4)$ , and x-intercept point $(2,0)$ .

Example

Suppose salami and sausage cost €6 and €3 per kilogram, and we wish to buy €12 worth. How much of each can we purchase? Letting x and y be the weights of salami and sausage, the total cost is: $6x+3y=12$ . Solving for y gives the point-slope form $y=-2x+4$ , as above. That is, if we first choose the amount of salami x, the amount of sausage can be computed as a function $y=f(x)=-2x+4$ . Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos: $f(x{+}1)=f(x)-2$ , and the slope is −2. The y-intercept point $(x,y)=(0,4)$ corresponds to buying only 4kg of sausage; while the x-intercept point $(x,y)=(2,0)$ corresponds to buying only 2kg of salami.

Note that we could instead have chosen y as the independent variable, and computed x as a linear function of it: $x=-{\tfrac {1}{2}}y+2$ . Also, the graph includes points with negative values of x or y, which have no meaning in terms of the original variables (unless we imagine selling meat to the butcher). Thus we should restrict our function to the domain $0\leq x\leq 2$ (or $0\leq y\leq 4$ ).

Relationship with other classes of functions

If the coefficient of the variable is not zero ( $a \neq 0$ ), then a linear function is represented by a degree 1 polynomial (also called a linear polynomial), otherwise it is a constant function – also a polynomial function, but of zero degree.

A straight line, when drawn in a different kind of coordinate system may represent other functions.

For example, it may represent an exponential function when its values are expressed in the logarithmic scale. It means that when $log (g (x))$ is a linear function of $x$ , the function $g$ is exponential. With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function.

If both arguments and values of a function are in the logarithmic scale (i.e., when $log (y)$ is a linear function of $log (x)$ ), then the straight line represents a power law:

\log _{r}y=a\log _{r}x+b\quad \Rightarrow \quad y=r^{b}\cdot x^{a}

On the other hand, the graph of a linear function in terms of polar coordinates:

r=f(\varphi )=a\varphi +b

is an Archimedean spiral if $a\neq 0$ and a circle otherwise.

Notes

^ Stewart 2012, p. 23
^ Stewart 2012, p. 24
^ Swokowski 1983, p. 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
Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Boston: Prindle, Weber & Schmidt, ISBN 0871503417

External links

[1] Stewart 2012, p. 23

[2] Stewart 2012, p. 24

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

[1]

[2]

[3]

v t e Polynomials and polynomial functions
By degree	Zero polynomial (degree undefined or −1 or −∞) Constant function (0) Linear function (1) Linear equation Quadratic function (2) Quadratic equation Cubic function (3) Cubic equation Quartic function (4) Quartic equation Quintic function (5) Sextic equation (6) Septic equation (7)
By properties	Univariate Bivariate Multivariate Monomial Binomial Trinomial Irreducible Square-free Homogeneous Quasi-homogeneous
Tools and algorithms	Factorization Greatest common divisor Division Horner's method of evaluation Polynomial identity testing Resultant Discriminant Gröbner basis