linear function

Examples of linear function in the following topics:

• Zeroes of Linear Functions

  • A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
  • The graph of a linear function is a straight line.
  • Linear functions can have none, one, or infinitely many zeros.  
  • To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
  • The zero from solving the linear function above graphically must match solving the same function algebraically.

• Linear and Quadratic Functions

  • Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
  • Linear and quadratic functions make lines and parabola, respectively, when graphed.
  • In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
  • Linear functions may be confused with affine functions.
  • Linear functions form the basis of linear algebra.

• What is a Linear Function?

  • Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
  • A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
  • It is linear: the exponent of the $x$ term is a one (first power), and it follows the definition of a function: for each input ($x$) there is exactly one output ($y$).  
  • The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.  
  • Identify what makes a function linear and the characteristics of a linear function

• Linear Approximation

  • A linear approximation is an approximation of a general function using a linear function.
  • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
  • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
  • If one were to take an infinitesimally small step size for $a$, the linear approximation would exactly match the function.
  • Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix.

• What is a Quadratic Function?

  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • Quadratic equations are different than linear functions in a few key ways.
  • Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope).
  • With a linear function, each input has an individual, unique output (assuming the output is not a constant).  
  • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

• Second-Order Linear Equations

  • A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • Linear differential equations are of the form $Ly = f$, where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and the right hand side $f$ is a given function of the same nature as $y$ (called the source term).
  • The linear operator $L$ may be considered to be of the form:
  • where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.

• Linear Equations

  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than $1$) or other function of a variable, equations involving terms such as $xy$, $x^2$, $y^{\frac{1}{3}}$, and $\sin x$ are nonlinear.
  • Linear differential equations are of the form:
  • where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and $f$ is a given function of the same nature as y (called the source term).
  • For a function dependent on time, we may write the equation more expressly as:

• The Equation of a Line

  • In statistics, linear regression can be used to fit a predictive model to an observed data set of $y$ and $x$ values.
  • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
  • Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.

• Linear Equations and Their Applications

  • Linear equations are those with one or more variables of the first order.
  • There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
  • Linear equations can therefore be expressed in general (standard) form as:
  • If the drivers want to designate a meeting point, they can algebraically find the point of intersection of the two functions, as seen in .
  • If the drivers want to designate a meeting point, they can algebraically find the point of intersection of the two functions.

• Nonhomogeneous Linear Equations

  • In the previous atom, we learned that a second-order linear differential equation has the form:
  • where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • Phenomena such as heat transfer can be described using nonhomogeneous second-order linear differential equations.
  • Identify when a second-order linear differential equation can be solved analytically