Linear function

Not to be confused with Linear equation.

Linear function with non-zero slope. (The line here is generic. See examples with concrete values below.)

There are several distinct definitions of a linear function. In basic mathematics, a linear function is a function whose graph is a straight line in the plane.. An example is: f(x)=2x–6. In higher mathematics, a linear function often refers to a linear mapping f between vector spaces such that f(ax+by)=af(x)+bf(y) for every x and y in the domain of f and every a and b in ℝ. In this case, a function whose graph is a straight line is called an affine function.

Linear Function as Line

Formally, a linear function f(x) is a real-valued function of one real-valued argument x, f:R→R such that the graph of f(x) is a line. Usually we write y(x) or just y in place of f(x).

There are three main forms of linear functions: standard, slope-intercept and parametric.

Slope-intercept form of a line (unique)

Slope-intercept form

The slope-intercept or point-slope or explicit form of a linear equation is  $y(x)=mx+b$  or  $y=mx+b$ . This form has 2 variables x and у and 2 constants m and b.

• The letters m and b are constants. Before working with a linear function, we replace m and b with actual real numbers.
• The letters x and y are variables.. The variable x is called the independent variable or argument. It is the input value. The variable y is called the dependent variable or function. It is the output value.
• The domain is R so any real number x can be input or substituted into a linear function. The function will then output the corresponding value for y.

• Horizontal lines are included. In this case, m=0 and y=b. Since b is a real number, this is a constant function. So a constant function is also a linear function.
• Vertical lines are never included. A vertical line is not a function. A vertical line does not pass the vertical line test. (A vertical line is defined by the equation: x=b where b is a real number.)
• Slanted lines are included. In this case, m≠0.

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• A linear function is a polynomial function of first or zero degree in one variable х .
• The slope-intercept form is unique. If we change the value of m or b, we get a different line.
• The constant term is b. If we substitute x=0 into the function, we get y=b. So the number b is the y-intercept and the line crosses the у-axis at the point (0,b).
• If m≠0, the number ^–b/_m is the x-intercept or root or zero and (^–b/_m,0) is the point where the line crosses the х-axis. Here, the value of the function is zero.
• The coefficient m of x is called the slope or gradient of the line. The slope determines the rate of change of the line so it determines both the "direction" and "steepness" of the line.
  • For each line, the slope is constant so the rate of change is always the same.
  • If the slope of a line is m and (х,у) is any point on the line, we must have that the point (х+1, y+m) is also on the line.
  • The sign of m determines the direction. If m>0 then the linear function is monotonely increasing (put your money in this bank!); if m&lt0 then the function monotonely decreases.
  • The absolute value of m determines the steepness. If |m|<1 then the slope is gentle; if |m|>1 then the slope is steep.

Example: y–2x+4. The slope is m= –2 and the y-intercept is b=4 or the point (0,4). Substituting y=0 and solving for x, we get 0=–2x+4 so x=2 is the root of this linear function and the point (2,0) is the x-intercept. Since the slope is m = –2, for each change in х of 1 (to the right), the value of у changes -2 (goes down).

• The graph of a line is determined by two points. To graph a linear function, we can substitute two different values for x into the function and solve for the corresponding y-values. We graph these two points. Using a straightedge, we draw the line through these two points extending it past both points.

Example: y–2x+4. Substituting x=0 we get y=4 (this is the y-intercept) and thus the point (0,4). Substituting x=1, we get y=2 and thus the point (1,2). Plot these points and draw the line. (Notice that the second point is one unit to the right and two units down from the first point. This happens for every point on this line because the slope is m= –2)

• A non-constant linear function is bijective. It will output every real number for exactly one input value.

Example: y–2x+4. Suppose y= –1. We substitute y(x)=–1 and get: –1= -x+2 or x=–3. This is the only solution.

Standard form of a line (non-unique). The unique slope-intercept form of this line is y=1.5x–0.5

Standard form

   $Ax+By=C,\,B\neq 0$.

This form is used mainly in geometry and in systems of linear equations. The standard form has 2 variables x and у and 3 constants A, B, and C that are replaced by real numbers before working. For example, in the linear function 3x–2y=1, the constants are A=3, B=–2 and C=1.

This form is sometimes written as: $Ax+By+C=0$, but this is equivalent since we always replace the constants by real numbers before working. So instead of writing 3x-2y=1, we write the equivalent equation 3x-2y-1=0.

In either form of the general form, the constants A, B, and C are not uniquely determined. If we multiply them by a factor k, these coefficients change, but the line is still the same. The two equations are linearly dependent and the lines are coincident.

Example: The lines 3x–2y=1 and 6x–4y=2 are the same line. Here the factor is: k=2. The unique slope-intercept form of this line is: y=1.5x–0.5 (solve either equation for y).

Vector-parametric form of a line (non-unique). The unique slope-intercept form of this line is y=1.5x+2.5

Vector-Parametric form

Parametric form: $\left\{{\begin{array}{*{20}{l}}{x(t)={x_{1}}+{a_{1}}t}\\{y(t)={y_{1}}+{a_{2}}t}\end{array}}\right.\,{a_{1}\neq 0}\,\,t\in \mathbb {R}$    or

Vector form: ${\mathbf {X} }=({x_{1}},{y_{1}})+t({a_{1}},{a_{2}})$  or  $r(t)=<x_{1}+a_{1}t,\,y_{1}+a_{2}t>,\,{a_{1}\neq 0}\,\,t\in \mathbb {R}$  .

The parametric or vector or vector-parametric form has 1 parameter t, 2 variables x and у, and 4 constants а₁, а₂, x₁, and y₁. The coefficients а₁, а₂, x₁, and y₁ are not uniquely determined. The line passes through the points А=(x₁,y₁) and B=(x₁+a₁,y₁+a₂) so that taking taking any other points or even just reversing the order ot the points will result in different constants for the same line.

• The parameter t is not visible on the graph.
• Engineers usually use the letter t for the parameter. Mathematicians often use the Greek letter λ. It does not matter.
• The vector-parametric form of a line extends naturally to lines in 3d and higher dimensional spaces.

Example: X=(–1,1)+t(2,3) is a line in vector form. Here: a₁=2, a₂=3, b₁=–1 and b₂=1. The line passes through the points (x₁,y₁)=(–1,1) and (x₁+a₁,y₁+a₂)=(1,4). The corresponding parametric form of this line is: x(t)= –1+2t, y(t)=1+3t. The unique slope-intercept form of this line is: y(x)=1.5x+2.5   (solve the first equation for t and substitute this result into the second equation).

Derivative of a linear function

In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values. A linear function has a constant rate of change. This rate of change is the slope m. So m is the derivative. This is often written:

$(mx+b)^{\prime }=m$

Example: y=–2x+4. Here m= –2 and so y′=–2.

Linear function vs. Linear equation

Often, the terms linear equation and linear function are confused. The word linear in linear equation means that all terms with variables are first degree. A linear equation can have 1, 2, 3, or more variables. So a linear equation is a linear function only if it has exactly 2 variables. (A linear equation in one variable is a point on the number line and a linear equation in 3 variables is a plane in 3d space.)

Notation

Many countries and disciplines use different letters and ordering for the different forms.

In many countries, a linear functions is often written as $y=ax+b$ where a is the slope and b is the y-intercept.

In business and economics, a linear function is often written as $y=a+bx$ where a is the y-intercept and b is the slope.

See also

• Line
• Polynomial
• Constant function
• Quadratic function
• Linear equation

Other websites

• http://www.mathopenref.com/linearexplorer.html (interactive)
• http://www.shodor.org/interactivate/activities/SlopeSlider/ (interactive)
• http://cs.fit.edu/~wds/classes/cse5255/thesis/lineEqn/lineEqn.html (equations)
• http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml (interactive)
• http://www.columbia.edu/itc/sipa/math/linear.html (examples from economics)
• http://www.tcd.ie/Economics/staff/ppwalsh/topic1.ppt (examples from economics)