Examples of linear function in the following topics:
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- 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.
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- 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.
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- 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
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- 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.
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- 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.
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- 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.
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- 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:
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- 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.
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- 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.
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- 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