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Linear function (calculus)

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 (in Cartesian coordinates with uniform scales) is a line in the plane.[1]
The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input.

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:[2]

.
Such a function is called linear because itsgraph, the set of all pointsin theCartesian plane, is aline. The coefficient a is called the slope of the function and of the line (see below).
If the slope is, this is a constant functiondefining a horizontal line, which some authors exclude from the class of linear functions.[3] With this definition, the degree of a linear polynomial would be exactly one, its graph a diagonal line neither vertical nor horizontal. However, we will not requirein this article, so constant functions will be considered linear.
The naturaldomainof a linear function, the set of allowed input values for x, is the entire set ofreal numbers,. One can also consider such functions with x in an arbitraryfield, taking the coefficients a,b in that field.
The graphis a non-vertical line having exactly one intersection with the y-axis, its y-intercept point. The y-intercept valueis also called the initial value of. If, the graph is a non-horizontal line having exactly one intersection with the x-axis, the x-intercept point. The x-intercept value, the solution of the equation, is also called the root orzeroof.

Slope

The slope of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run). If the line is the graph of the linear function f(x) = ax + b, this slope is given by the constant a.

The slope measures the constant rate of change ofper unit change in x: whenever the inputxis increased by one unit, the output changes byaunits:, and more generallyfor any number. If the slope is positive,, then the functionis increasing; if, thenis decreasing
Incalculus, the derivative of a general function measures its rate of change. A linear functionhas a constant rate of change equal to its slopea, so its derivative is the constant function.
The fundamental idea of differential calculus is that anysmoothfunction(not necessarily linear) can be closelyapproximatednear a given pointby a unique linear function. Thederivativeis the slope of this linear function, and the approximation is:for. The graph of the linear approximation is thetangent lineof the graphat the point. The derivative slopegenerally varies with the point c. Linear functions can be characterized as the only real functions whose derivative is constant: iffor all x, thenfor.

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

A given linear functioncan be written in several standard formulas displaying its various properties. The simplest is the slope-intercept form:
,
from which one can immediately see the slope a and the initial value, which is the y-intercept of the graph.
Given a slope a and one known value, we write the point-slope form:
.
In graphical terms, this gives the linewith slope a passing through the point.
The two-point form starts with two known valuesand. One computes the slopeand inserts this into the point-slope form:
.
Its graphis the unique line passing through the points. The equationmay also be written to emphasize the constant slope:
.

Relationship with linear equations

Linear functions commonly arise from practical problems involving variableswith a linear relationship, that is, obeying alinear equation. If, one can solve this equation for y, obtaining
where we denoteand. That is, one may consider y as a dependent variable (output) obtained from the independent variable (input) x via a linear function:. In the xy-coordinate plane, the possible values ofform a line, the graph of the function. Ifin the original equation, the resulting lineis vertical, and cannot be written as.
The features of the graphcan be interpreted in terms of the variables x and y. The y-intercept is the initial valueat. 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,. Negative slope a indicates a decrease in y for each increase in x.
For example, the linear functionhas slope, y-intercept point, and x-intercept point.

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:. Solving for y gives the point-slope form, as above. That is, if we first choose the amount of salami x, the amount of sausage can be computed as a function. Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos:, and the slope is −2. The y-intercept pointcorresponds to buying only 4kg of sausage; while the x-intercept pointcorresponds to buying only 2kg of salami.
Note that 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 functionto the domain.
Also, we could choose y as the independent variable, and compute x by theinverselinear function:over the domain.

Relationship with other classes of functions

If the coefficient of the variable is not zero (a ≠ 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:

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

is anArchimedean spiralifand acircleotherwise.

References

[1]
Citation Linkopenlibrary.orgStewart 2012, p. 23
Sep 27, 2019, 7:59 PM
[2]
Citation Linkopenlibrary.orgStewart 2012, p. 24
Sep 27, 2019, 7:59 PM
[3]
Citation Linkopenlibrary.orgSwokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Boston: Prindle, Weber & Schmidt, ISBN 0871503417, p. 34
Sep 27, 2019, 7:59 PM
[4]
Citation Linkweb.archive.orghttps://web.archive.org/web/20130524101825/http://www.math.okstate.edu/~noell/ebsm/linear.html
Sep 27, 2019, 7:59 PM
[5]
Citation Linkwww.corestandards.orghttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Sep 27, 2019, 7:59 PM
[6]
Citation Linkarchive.orgCalculus with analytic geometry
Sep 27, 2019, 7:59 PM
[7]
Citation Linkweb.archive.orghttps://web.archive.org/web/20130524101825/http://www.math.okstate.edu/~noell/ebsm/linear.html
Sep 27, 2019, 7:59 PM
[8]
Citation Linkwww.corestandards.orghttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Sep 27, 2019, 7:59 PM
[9]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 27, 2019, 7:59 PM