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.

Properties of Linear Functions

• A linear function is a polynomial function of first degree with one independent variable х, i.e.

  ${\displaystyle y(x)=ax+b}$   (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: ${\displaystyle y(x)=2x-1}$. 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  ${\displaystyle \Re }$  (all real numbers). This means that any real number can be substituted for x. (Of course, the value of y depends on the substituted value 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).

160px 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 ax+b=0.
  The zero is x=-b⁄a.

• There are three standard forms for linear functions.
  • General form
  • Slope-intercept form
  • Vector-parametric form

General Form

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

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.

The general form of a line is a linear equation; the opposite is not necessarily true.

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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

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

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)=-2х+4
• The slope-intercept form is unique. That is, if the value of either or both of the coefficient letters a and b are changes, the result is a different line!
• 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 because the y-axis is the line where x=0. Substituting x=0 into the linear function y(x)=ax+b, it follows that: y(0)=a•0+b=b. This means that the point (0,b) is both a point on the line and a point on the y-axis. So it is the point where the line intersects the y-axis.
  • The number -b⁄а is the [root] or [zero] of the function. It is the x-value at which the line intersects the x-axis. This is because the x-axis is the line where y=0. Substituting y=0 into the linear function and solving (backwards) for x, it follows that:
           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.

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Example: ${\displaystyle y(x)=-2x+4}$    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

Parametri: ${\displaystyle \left\{{\begin{array}{*{20}{l}}{x(t)={b_{1}}+{a_{1}}t}\\{y(t)={b_{2}}+{a_{2}}t}\end{array}}\right.}$     or Vector: ${\displaystyle {X}=({b_{1}},{b_{2}})+t({a_{1}},{a_{2}})}$    where    ${\displaystyle {a_{1}\neq 0}}$   and  ${\displaystyle {a_{2}\neq 0}}$ .

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.

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Example: ${\displaystyle {X}=({-1},{1})+t({2},{3})}$    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+a2)=(1,4)  These are the points where t=0 and t=1 (t is not visible on the graph!). The parametric form of this line is: ${\displaystyle \left\{{\begin{array}{*{20}{l}}{x(t)={-1}+{2}t}\\{y(t)={1}+{3}t}\end{array}}\right.}$    The slope-intercept form of this line is: y(x)=1,5x+2,5   (solve the first parametric equation for t and substitute into the second). One general form of this line is: -3x+2y=5.

References

• http://www.columbia.edu/itc/sipa/math/linear.html
• http://www.math.okstate.edu/~noell/ebsm/linear.html
• http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf#page=55