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 $x$ , i.e. $y(x)=ax+b$ .^[2] Here $x$ is the independent variable and $y$ is the dependent variable.

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

The set of points $(x, y (x))$ is the line that is the graph of the function. Because 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. Because $y (x)$ is a function, the line cannot be vertical.

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

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.

There are three standard forms for 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 a linear function is an equation of the form

Ax+By=C

where $B\neq 0$ .

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. For example, $3 x - 2 y = 1$ and $9 x - 6 y = 3$ are general forms of the same line.

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

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

[1]

[2]

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