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Linear function

A linear function is a mathematical function of one real variable whose graph is a straight line, expressed in the standard form f(x) = mx + b, where m is the slope representing the constant rate of change of the output with respect to the input, and b is the y-intercept denoting the value of the function when x = 0.^[1]^[2] This form captures relationships where the dependent variable changes proportionally to the independent variable, making it a cornerstone of introductory algebra and calculus.^[3] In higher mathematics, the term "linear function" has a stricter definition: a function f: V \to W between vector spaces that preserves addition and scalar multiplication, satisfying f(x + y) = f(x) + f(y) and f(cx) = c f(x) for all scalars c and vectors x, y.^[4] Under this definition, which implies f(0) = 0, the function takes the form f(x) = mx and passes through the origin; the more general equation f(x) = mx + b with b \neq 0 is technically an affine function, though commonly called linear in elementary contexts due to its linear graph.^[4]^[3] This distinction is crucial in fields like linear algebra, where linearity ensures compatibility with vector space operations. Key properties of linear functions include their constant slope, which determines steepness and direction (positive for increasing, negative for decreasing, zero for horizontal lines), and their ability to be graphed by plotting two points or using point-slope form y - y_1 = m(x - x_1).^[1]^[2] They model numerous real-world phenomena with uniform rates, such as distance traveled at constant speed in physics, cost accumulation in economics, population growth approximations in biology, and depreciation in business.^[5]^[1]^[6] Linear functions also form the basis for solving systems of equations, optimization problems, and approximations in more complex analyses like regression.^[7]^[8]

Definition and Forms

General Definition

A linear function is a mathematical relation f: \mathbb{R} \to \mathbb{R} that maps each real number input to a unique real number output, expressed in the form f(x) = mx + b, where m and b are real constants, with m representing the slope and b the y-intercept.^[9] This form assumes a basic understanding of functions as rules assigning outputs to inputs within a specified domain and range; here, the domain comprises all real numbers \mathbb{R}, as the expression is defined for every real x, while the range is \mathbb{R} if m \neq 0, or the singleton \{b\} if m = 0.^[10] The linearity of such a function arises from its preservation of proportionality in the relationship between input and output, adjusted by a constant shift: the output scales directly with the input by the factor m, and adding b translates the result vertically.^[9] For instance, when the input x = 0, the function yields f(0) = b, illustrating the intercept's role as the baseline value before any scaling effect.^[2] This structure ensures the function models straight-line behaviors in one dimension, distinguishing it from nonlinear relations that curve or bend. The origins of linear functions trace to 17th-century analytic geometry, pioneered independently by René Descartes and Pierre de Fermat, who formalized straight-line relations through algebraic equations on coordinate planes.^[11] Descartes's La Géométrie (1637) and Fermat's posthumous works integrated algebra with geometry, enabling the representation of lines as proportional expressions that underpin modern linear modeling.^[12]

Standard Forms

Linear functions can be expressed in several standard algebraic forms, each suited to specific contexts such as graphing, geometric applications, or solving systems of equations. These forms are derived from the general definition of a linear function f(x) = mx + b, where m represents the slope and b the y-intercept, and provide practical tools for manipulation and analysis.^[13]^[14] The slope-intercept form is y = mx + b, obtained directly from the general definition by solving for y. This form is particularly useful when the y-intercept is known, as it immediately reveals the slope m (the rate of change) and allows quick graphing by identifying the y-intercept and using the slope to find additional points. For instance, in modeling real-world scenarios like cost functions where the fixed cost (y-intercept) is given, this form simplifies parameter identification.^[13]^[14] The point-slope form is y - y_1 = m(x - x_1), where (x_1, y_1) is a known point on the line and m is the slope. To derive this from two points, say (x_1, y_1) and (x_2, y_2), first compute the slope m = \frac{y_2 - y_1}{x_2 - x_1}, then substitute one point into the form. This is advantageous in geometric problems, such as finding equations of lines passing through specific coordinates, as it avoids initial intercept calculations.^[15]^[16] The general form is ax + by + c = 0, where a, b, and c are constants with a and b not both zero. This compact representation is equivalent to other forms and facilitates comparisons in systems of equations. The standard form for lines, a variant, is Ax + By = C, where A, B, and C are integers, A \geq 0, and the coefficients have no common factors other than 1; it is ideal for integer-based computations and solving simultaneous equations.^[13]^[17] Conversions between these forms involve algebraic rearrangements. To convert from slope-intercept y = mx + b to standard form Ax + By = C, subtract mx from both sides to get -mx + y = b, then multiply by -1 if needed to make the x-coefficient positive: for y = 2x - 3, this yields -2x + y = -3, or $2x - y = 3.^[14]^[17] From point-slope to slope-intercept, distribute the slope term and simplify: starting with y - 4 = 3(x - 1), expand to y - 4 = 3x - 3, add 4 to both sides for y = 3x + 1.^[13] To convert from standard form Ax + By = C to slope-intercept, isolate y by subtracting Ax and dividing by B (assuming B \neq 0): for $3x + 4y = 12, subtract $3x to get $4y = -3x + 12, then divide by 4 for y = -\frac{3}{4}x + 3. From general form ax + by + c = 0 to slope-intercept, first rewrite as ax + by = -c, then solve similarly: for $2x - 3y + 6 = 0, move the constant to get $2x - 3y = -6, then -3y = -2x - 6, divide by -3 for y = \frac{2}{3}x + 2.^[13]^[17]

Properties

Basic Properties

A linear function of the form f(x) = mx + b, where m and b are real constants, exhibits a constant slope m between any two distinct points in its domain. Specifically, for any x_1 \neq x_2, the difference quotient is \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{(m x_2 + b) - (m x_1 + b)}{x_2 - x_1} = m, demonstrating that the rate of change remains invariant regardless of the chosen points.^[18]^[19] Strictly speaking, functions satisfying additivity f(x + y) = f(x) + f(y) and homogeneity f(cx) = c f(x) for all real scalars c and inputs x, y are linear only if b = 0, yielding f(x) = mx with f(0) = 0; the general form f(x) = mx + b is affine. For f(x) = mx + b, f(x + y) = m(x + y) + b = mx + my + b, while f(x) + f(y) = mx + b + my + b = mx + my + 2b, so additivity fails unless b = 0.^[4]^[20] Such functions are invertible if and only if m \neq 0, in which case they are both one-to-one (injective) and onto the real numbers (surjective), with the explicit inverse given by f^{-1}(y) = \frac{y - b}{m}.^[21]^[22] Linear functions are continuous at every point in their domain, as they are polynomials of degree at most one, and differentiable everywhere with constant derivative f'(x) = m.^[23]^[19] Unlike piecewise linear functions, which consist of affine segments over disjoint intervals and may exhibit discontinuities or varying slopes, true linear functions maintain a single affine expression across their entire domain without breaks.^[24]

Characteristic Behaviors

Linear functions exhibit a uniform rate of change across their domain, meaning the output increases or decreases by a fixed amount for every unit increase in the input variable. This constant rate is quantified by the slope m in the standard form f(x) = mx + b, where the difference f(x + h) - f(x) = mh holds for any h \neq 0. For example, the function f(x) = 2x + 1 increases by 2 units in output for each 1-unit increase in x, as f(2) = 5 and f(3) = 7.^[7]^[25]^[26] This behavior reflects proportionality with an offset: the output scales directly with the input by the factor m, but is shifted by the constant b, distinguishing linear functions from strictly proportional ones where b = 0. In affine terms, linear functions (broadly including the intercept) can be viewed as a linear transformation plus a constant offset, ensuring the scaling property persists regardless of the shift.^[27]^[28] The range of a linear function is typically unbounded over the real numbers, extending to positive and negative infinity unless the domain is restricted, with no horizontal or vertical asymptotes due to the absence of poles or horizontal leveling. For instance, f(x) = 2x + 1 takes all real values as x varies over \mathbb{R}.^[29]^[30] The composition of two linear functions remains linear, preserving the form f(g(x)) = m_1(m_2 x + b_2) + b_1 = (m_1 m_2) x + (m_1 b_2 + b_1). As a simple example, if f(x) = 3x + 1 and g(x) = 2x - 4, then f(g(x)) = 3(2x - 4) + 1 = 6x - 11, which is linear with slope 6.^[31]^[32] Limiting cases include the horizontal line when m = 0, yielding a constant function f(x) = b with zero rate of change and a bounded range equal to \{b\}. Vertical lines, corresponding to undefined slope, are excluded as they fail to define a function under the standard definition, as a single input maps to multiple outputs.^[33]^[34]^[22]

Graphical Representation

Graph Characteristics

The graph of a linear function, expressed in the form f(x) = mx + b where m and b are constants, is always a straight line in the Cartesian plane.^[3]^[35] This straight-line nature arises because the function maintains a constant rate of change, ensuring that the plot connects any two points on the line without curvature.^[36] The y-intercept occurs where the graph crosses the y-axis, at the point (0, b), which represents the value of the function when x = 0.^[35]^[36] The x-intercept, if it exists (when m \neq 0), is found by setting f(x) = 0 and solving for x, yielding x = -b/m, marking where the line crosses the x-axis.^[35] These intercepts are significant for graphing, as plotting them allows a straight line to be drawn through the points, and they provide key insights into the function's behavior, such as initial values or zero crossings in applied contexts.^[36] The slope m is interpreted geometrically as the ratio of rise (change in y) to run (change in x), quantifying the line's steepness and direction.^[33] A positive slope indicates the line rises from left to right, a negative slope shows it falls, a zero slope results in a horizontal line parallel to the x-axis, and an undefined (infinite) slope corresponds to a vertical line (though typically excluded from standard linear functions).^[35] Lines with the same slope m are parallel, never intersecting, as they maintain equal rates of change.^[37] Perpendicular lines have slopes m_1 and m_2 satisfying m_1 m_2 = -1, where one slope is the negative reciprocal of the other; for example, a line with slope 2 is perpendicular to one with slope -1/2.^[37] The graph of a linear function extends infinitely in both directions along the x-axis, reflecting its domain of all real numbers unless explicitly restricted, visualizing an unbounded straight path across the plane.^[35]

Transformations

Transformations of the graphs of linear functions preserve the straight-line nature while altering position, orientation, or apparent scale. These operations include translations, reflections, scalings, and rotations, each corresponding to modifications in the slope-intercept form f(x) = mx + b, where m is the slope and b is the y-intercept. Such transformations are fundamental in understanding how parameters shift under geometric changes.^[38]

Translations

Vertical translations shift the graph up or down without changing its slope. Adding a constant k to the function yields g(x) = f(x) + k = mx + (b + k), increasing the y-intercept by k if k > 0 or decreasing it if k < 0. For example, the line y = 2x + 1 translated up by 3 units becomes y = 2x + 4.^[38] Horizontal translations move the graph left or right, affecting the effective intercept but not the slope directly. The form g(x) = f(x - h) = m(x - h) + b = mx + (b - mh) shifts the graph right by h units if h > 0, altering the x-intercept from -b/m to -b/m + h. For instance, shifting y = 2x + 1 right by 1 unit gives y = 2(x - 1) + 1 = 2x - 1.^[39]

Reflections

Reflections flip the graph over an axis, negating the slope or intercept. Reflection over the x-axis produces g(x) = -f(x) = -mx - b, reversing the direction of the line and negating both slope and y-intercept. The line y = 2x + 1 becomes y = -2x - 1.^[38] Reflection over the y-axis gives g(x) = f(-x) = -mx + b, negating only the slope while preserving the y-intercept. Thus, y = 2x + 1 transforms to y = -2x + 1. These operations change the line's inclination but maintain its linearity.^[39]

Scaling

Scalings adjust the steepness or spread of the graph. A vertical stretch by factor c > 0 (or compression if $0 < c < 1) results in g(x) = c f(x) = cmx + cb, multiplying both slope and y-intercept by c. For y = 2x + 1 with c = 3, the new equation is y = 6x + 3, steepening the line.^[39] A horizontal stretch by factor d > 0 (or compression if $0 < d < 1) uses g(x) = f(x/d) = m(x/d) + b = (m/d)x + b, dividing the slope by d while leaving the y-intercept unchanged. Applying d = 2 to y = 2x + 1 yields y = x + 1, flattening the line. Note that horizontal scaling affects the slope but not the vertical position at x = 0.^[40]

Rotations

Rotations alter the orientation of the line around a point, typically the origin, changing its slope based on the rotation angle. The slope m equals \tan \theta, where \theta is the angle the line makes with the positive x-axis./01%3A_Sections/1.32%3A_Slope) Rotating by an angle \phi counterclockwise adjusts the angle to \theta + \phi, so the new slope is m' = \tan(\theta + \phi) = \frac{m + \tan \phi}{1 - m \tan \phi}, assuming $1 - m \tan \phi \neq 0 to avoid undefined cases like 90-degree rotations./07%3A_Trigonometric_Identities_and_Equations/7.02%3A_Sum_and_Difference_Identities) For a line not passing through the origin, translation to the origin, rotation, and translation back are required, complicating the intercept. The resulting equation remains linear with the updated slope.^[41]

Combined Transformations

Multiple transformations are applied sequentially, with order affecting the outcome since operations like scaling and translation do not commute. For example, vertically scaling then translating differs from translating then scaling, as the constant shift is multiplied in the former. Reflections and rotations also interact non-commutatively with shifts. These compositions can be represented using affine transformations, which combine linear parts with translations, though explicit matrix forms are beyond basic scalar descriptions here.^[39]

Linear Functions in Multiple Variables

Functions of Several Variables

A linear function of several variables is a function f: \mathbb{R}^n \to \mathbb{R} that can be expressed as f(\mathbf{x}) = a_1 x_1 + a_2 x_2 + \dots + a_n x_n + b, where \mathbf{x} = (x_1, x_2, \dots, x_n) is the input vector, a_1, \dots, a_n are constant coefficients, and b is a constant term.^[42] In vector notation, this simplifies to f(\mathbf{x}) = \vec{a} \cdot \mathbf{x} + b, where \vec{a} = (a_1, \dots, a_n) is the coefficient vector.^[43] This form extends the single-variable linear function f(x) = ax + b to higher dimensions while preserving the property of constant rates of change along each coordinate direction.^[18] The graph of such a function, considered as a subset of \mathbb{R}^{n+1} with coordinates (\mathbf{x}, f(\mathbf{x})), forms a hyperplane.^[44] Specifically, the equation z = \vec{a} \cdot \mathbf{x} + b rearranges to \vec{a} \cdot \mathbf{x} - z + b = 0, defining an affine hyperplane in \mathbb{R}^{n+1}. The normal vector to this hyperplane is (\vec{a}, -1) = (a_1, \dots, a_n, -1), which is perpendicular to every direction lying within the hyperplane.^[45] For n=2, this hyperplane reduces to a plane in three-dimensional space, as seen in the standard form z = a_1 x + a_2 y + b.^[18] The partial derivatives of a linear function are constant and equal to the respective coefficients. For f(\mathbf{x}) = \sum_{i=1}^n a_i x_i + b, the partial derivative with respect to x_i is \frac{\partial f}{\partial x_i} = a_i for each i = 1, \dots, n.^[46] These constants reflect the uniform slope of the function along each variable's axis, independent of the values of the other variables. Higher-order partial derivatives are all zero, underscoring the function's linearity.^[47] Level sets of a linear function, where f(\mathbf{x}) = c for some constant c, consist of parallel hyperplanes in \mathbb{R}^n. The equation simplifies to \vec{a} \cdot \mathbf{x} = c - b, which defines an (n-1)-dimensional affine hyperplane with normal vector \vec{a}.^[18] These level sets are equally spaced and parallel, as shifting c translates the hyperplane along the direction of \vec{a}. For n=2, they appear as parallel lines in the plane.^[48] An example is the linear regression model with two predictors, y = \beta_0 + \beta_1 x_1 + \beta_2 x_2, where \beta_0, \beta_1, \beta_2 are fixed parameters, modeling the scalar output y as a linear combination of inputs x_1 and x_2.^[49] The graph of this model is a plane in three-dimensional space.^[49]

Relation to Linear Maps

In linear algebra, a linear map, also known as a linear transformation, is a function T: V \to W between vector spaces V and W over the same field that preserves vector addition and scalar multiplication, satisfying T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) and T(c\mathbf{u}) = c T(\mathbf{u}) for all vectors \mathbf{u}, \mathbf{v} \in V and scalars c.^[50] This definition generalizes the concept of a linear function from the real line, where a function f(x) = mx (with m constant and no intercept) maps \mathbb{R} to \mathbb{R} while preserving these operations, but extends it to arbitrary vector spaces. In contrast, an affine map includes a translation term, such as f(\mathbf{x}) = A\mathbf{x} + \mathbf{b} where A is linear and \mathbf{b} \neq \mathbf{0}, which does not preserve the origin unless \mathbf{b} = \mathbf{0}.^[51] In finite-dimensional spaces, where \dim V = n and \dim W = m, every linear map T: V \to W can be represented by an m \times n matrix A with respect to chosen bases for V and W, such that T(\mathbf{x}) = A\mathbf{x} in coordinates. For instance, a linear function from \mathbb{R} to \mathbb{R} corresponds to multiplication by a scalar m, represented by the $1 \times 1 matrix ; an affine version like f(x) = mx + b (with b \neq 0) is not linear but can be viewed as a linear map in a higher-dimensional augmented space.^[52] The kernel of a linear map T, denoted \ker(T), is the set \{\mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0}\}, which forms a subspace of V; T is injective if and only if \ker(T) = \{\mathbf{0}\}.^[53] The image, or range, \operatorname{im}(T) = \{ T(\mathbf{v}) \mid \mathbf{v} \in V \}, is a subspace of W spanned by the columns of the matrix A representing T. For linear functions without a constant term (i.e., b = 0), these structures highlight injectivity and surjectivity properties directly.^[54] The matrix representation of a linear map depends on the choice of bases. For a linear map that is an endomorphism (V = W) with the same change-of-basis matrix S (whose columns are the new basis vectors expressed in the old basis coordinates) for both domain and codomain, the new matrix B satisfies B = S^{-1} A S.^[55] In general, for linear maps T: V → W, if Q is the change-of-basis matrix for the domain and P for the codomain (columns are new basis vectors in old coordinates), then B = P^{-1} A Q.^[56] This preserves the map's properties across bases. A linear map T: V \to W is an isomorphism if it is bijective, meaning it has an inverse that is also linear; this occurs precisely when V and W have the same finite dimension and the representing matrix is invertible.^[57] For example, in equal dimensions, a linear function f(x) = mx with m \neq 0 induces an isomorphism on \mathbb{R}, as its matrix is invertible.^[58]

Applications

In Everyday Contexts

Linear functions appear frequently in everyday scenarios where relationships between quantities change at a constant rate. One common example is in cost models for services like taxi fares, which typically include a fixed initial charge plus a variable fee based on distance traveled. For instance, a taxi fare can be modeled as f(d) = 3 + 0.5d, where d is the distance in miles and f(d) is the total cost in dollars, reflecting a $3 base fee and $0.50 per mile.^[59] This linear structure allows riders to predict expenses based on trip length, assuming uniform pricing. Temperature conversions between scales also rely on linear functions, providing a straightforward way to translate measurements across systems. The conversion from Celsius to Fahrenheit is given by f(C) = \frac{9}{5}C + 32, where C is the temperature in Celsius and f(C) is the equivalent in Fahrenheit.^[60] This formula originated from the Fahrenheit scale, proposed in 1724 by physicist Daniel Gabriel Fahrenheit to standardize thermometer readings using fixed points like the freezing of water at 32°F and human body temperature near 96°F.^[61] In transportation and motion, linear functions describe uniform movement, such as distance covered over time at constant speed. For an object starting from position s_0 and moving at velocity v, the position is s(t) = s_0 + vt, where t is time. This model applies to everyday travel, like a train departing a station, enabling predictions of arrival times based on steady progress. Budgeting often involves linear functions for tracking savings growth under simple accumulation rules. For example, if someone begins with $400 in savings and adds $75 weekly, the total saved after w weeks is s(w) = 400 + 75w.^[62] This linear progression helps individuals plan finances by forecasting balances without compounding interest. Proportional relationships, a subset of linear functions passing through the origin, model scenarios like work rates where output scales directly with input. A worker completing jobs at a constant rate of 2.5 per hour would finish j(h) = 2.5h jobs in h hours, assuming steady effort.^[63] Such models aid in scheduling tasks by estimating completion times from productivity rates.

In Mathematics and Science

In calculus, the antiderivative of a linear function f(x) = mx + b is straightforward, given by \int (mx + b) \, dx = \frac{m}{2}x^2 + bx + C, where C is the constant of integration.^[64] This computation exemplifies the integration techniques foundational to calculus, as linear functions yield quadratic antiderivatives that are easily evaluated.^[65] Linear functions also play a key role in illustrating the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f, then the definite integral \int_a^b f(x) \, dx = F(b) - F(a); for linear f, this directly quantifies net change over an interval through the theorem's evaluation property.^[65] In physics, linear functions model fundamental relationships in mechanics. Hooke's law, for instance, expresses the restoring force of a spring as F = -kx, where k is the spring constant and x is the displacement from equilibrium, assuming small deformations where the force is directly proportional to displacement.^[66] Similarly, in kinematics for motion with constant acceleration, velocity varies linearly with time according to v = u + at, where u is the initial velocity, a is acceleration, and t is time; this equation derives from the definition of acceleration as the rate of change of velocity.^[67] In economics, linear functions commonly represent supply and demand curves, with supply increasing linearly with price and demand decreasing, allowing equilibrium to be determined at their intersection point where quantity supplied equals quantity demanded.^[68] This intersection solves the system of linear equations for the market-clearing price and quantity, providing a foundational tool for analyzing market balance.^[69] In computer science, particularly computer graphics, linear interpolation blends between two points P_0 and P_1 using the formula f(t) = (1-t)P_0 + tP_1 for t \in [0,1], generating straight-line segments essential for rendering smooth transitions and parametric curves.^[70] In statistics, simple linear regression models the expected value of a response variable as a linear function of a predictor, E(y|x) = \beta_0 + \beta_1 x, under assumptions including linearity in parameters, independence of errors, homoscedasticity (constant error variance), and normality of errors for inference.^[71] The slope parameter \beta_1 is estimated via least squares as \hat{\beta_1} = \frac{\text{Cov}(x,y)}{\text{Var}(x)}, which minimizes the sum of squared residuals and captures the average change in y per unit change in x.^[72]

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Composition of linear functions. Proposition. Let f : V → W and g : W → U be linear functions. Their the composition g ◦ f : V → U is a linear function.
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[PDF] Vector Spaces and Linear Functions
The composition of linear functions is linear. Composition of linear functions corresponds to multiplication of their matrices: If L(y) = Ay and K(x) = Bx ...
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Tutorial 15: The Slope of a Line - West Texas A&M University
Jul 3, 2011 · Note that when a line is horizontal the slope is 0. Note that when the line is vertical the slope is undefined.
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[PDF] Constant and Linear Functions
Finally, we want to point out that, when m is 0, we simply have a constant function again. When m is positive, the line is sloped upward, and when m is negative ...
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Lesson 8: Functions and Their Graphs - EdTech Books
The graph of a linear function, f ( x ) = m x + b is a straight line that crosses the y -axis at the point ( 0 , b ) and has a slope of m . The illustrations ...Missing: characteristics | Show results with:characteristics
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### Summary of Graphing Linear Equations Content
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[PDF] Appendix C Lines and Linear Functions
Parallel and perpendicular Lines with equal slope are parallel. The slopes m₁ and m2 of perpendicular lines are negative reciprocals (i.e., m1m2 = -1). ...
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Algebra - Transformations - Pauls Online Math Notes
Nov 16, 2022 · In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the x and y-axis.Missing: stretches | Show results with:stretches
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1-07 Transformations of Functions
Translations and reflections moved the graph, but did not change its shape. Stretches and shrinks are transformations that do change the shape by stretching or ...
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1.5 - Shifting, Reflecting, and Stretching Graphs
There are two kinds of translations that we can do to a graph of a function. They are shifting and scaling. There are three if you count reflections, but ...Missing: rotations | Show results with:rotations
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https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_(OpenStax)/12%3A_Analytic_Geometry/12.04%3A_Rotation_of_Axes
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[PDF] 14.1 Functions of Several Variables
Oct 1, 2012 · A linear function is a function f (x) = ax + by + c. The graph of such a function is a plane. (). 14.1 Functions of Several Variables. October 1 ...
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Functions of Several Variables
We'll usually call the variables x and y, and call the function f(x,y). The domain is the set of pairs (x,y) that are ...
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[PDF] Linear Predictors, Part 1 - Duke Computer Science
Sep 20, 2021 · of all affine functions on Rd. The graph of an affine function on Rd is a hyperplane in Rd+1. For instance, when d = 1, the graph of y ...<|control11|><|separator|>
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Planes in ℝ3 - Ximera - The Ohio State University
The graph of such an equation, for , is called a hyperplane. The vector in whose components are the coefficients is orthogonal to every vector in the hyperplane ...
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Calculus III - Partial Derivatives - Pauls Online Math Notes
Nov 16, 2022 · Just as with functions of one variable we can have derivatives of all orders. We will be looking at higher order derivatives in a later section.
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2. Partial Derivatives | Multivariable Calculus - MIT OpenCourseWare
In this unit we will learn about derivatives of functions of several variables. Conceptually these derivatives are similar to those for functions of a single ...
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Calculus III - Functions of Several Variables - Pauls Online Math Notes
Nov 16, 2022 · The level curves of the function z=f(x,y) z = f ( x , y ) are two dimensional curves we get by setting z=k z = k , where k k is any number. So ...
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5.3 - The Multiple Linear Regression Model | STAT 501
For more than two predictors, the estimated regression equation yields a hyperplane. Interpretation of the Model Parameters. Each ...
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[PDF] Linear Transformations
◦ Reflecting happens if di is negative. We write Rotθ : R2 → R2 for the linear transformation which rotates vectors in R2 counter-clockwise through the angle θ ...
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[PDF] ∼ = Mm×n(F).
Defined the matrix [T] of a linear map with respect to a choice of bases. Remark 1. Linear vs affine. The function f : R → R, f(x) = mx + b is linear iff b = 0 ...
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[PDF] LINEAR TRANSFORMATIONS Math 21b, O. Knill
A transformation is also called a map. LINEAR TRANSFORMATION. A map T from Rn to Rm is called a linear transformation if there is a m × n matrix A such that.Missing: definition algebra
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[PDF] Lecture 13: Image and Kernel
Mar 2, 2011 · Also the kernel of a matrix A is a linear space. How do we compute the kernel? Just solve the linear system of equations Ax = 0. Form rref(A).
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[PDF] Homework 7: Image and Kernel
The kernel is the set of vectors x which satisfy Ax = 0. The image of a linear map x → Ax is the set of all vectors. Ax. The columns of A span the image of A.
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[PDF] Unit 5: Change of Coordinates
Since b is a basis, S is invertible and [v]B = S−1v. D. Theorem: If T(x) = Ax is a linear map and S is the matrix from a basis change, then B = S−1AS is ...
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[PDF] Representation theory of finite groups II: Linear algebra
Jul 8, 2015 · So all finite-dimensional vector spaces and linear maps between them become familiar objects from matrix algebra after choosing bases.
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16.3 Singular Transformations and Matrices
A vector that is mapped into the 0 vector by a transformation is said to be in the kernel of that transformation. Since the transformation is linear any linear ...Missing: algebra | Show results with:algebra<|control11|><|separator|>
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2.1 Linear Functions - Precalculus 2e | OpenStax
Dec 21, 2021 · The train's distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from ...
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Daniel Gabriel Fahrenheit and the Birth of Precision Thermometry
Gabriel Daniel Fahrenheit, inventor of the mercury thermometer and the Fahrenheit temperature scale. The teenage Fahrenheit, to his guardians' irritation, ...
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3.2 Linear Growth | Mathematics for the Liberal Arts Corequisite
She started with $400 and is now saving $75 per week from her part-time job. Let y = total amount saved, and x = the number of weeks since beginning to save.
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[PDF] 1.5 Linear functions and proportional models
Changes in input and output. Suppose y is a function of x. Then there is some rule that answers the question: What is the value of y for any given x?
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[PDF] 5.3 Fundamental Theorem of Calculus - CSUN
Fundamental Theorem of Calculus ». Example 2 suggests that the area function A for a linear function f is an antiderivative of f ; that is, A ' (x) = f (x).
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1.3 The Fundamental Theorem of Calculus
The form ∫ a b G ′ ( x ) d x = G ( b ) − G ( a ) of the fundamental theorem relates the rate of change of G ( x ) over the interval a ≤ x ≤ b to the net change ...
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Spring force and Hooke's law (article) | Khan Academy
This article discusses the force exerted by a spring and how to model and predict spring forces using Hooke's law.
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Equilibrium, Price, and Quantity | Introduction to Business
On a graph, the point where the supply curve (S) and the demand curve (D) intersect is the equilibrium. The equilibrium price is the only price where the ...
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3.4 – Demand, Supply, and Equilibrium - eCampusOntario Pressbooks
The equilibrium occurs where supply and demand curves intersect. Panel (a) depicts an increase in demand. Supply Curve (S1) is linear sloping upward from ...
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[PDF] Lecture 24: Bezier Curves and Surfaces
(2.5). Equation (2.5) is called linear interpolation because the curve P(t) interpolates the points P0,P1 with a straight line. You have encountered linear ...
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12.3 - Simple Linear Regression - STAT ONLINE
Recall, the equation for a simple linear regression line is y ^ = b 0 + b 1 x where b 0 is the y -intercept and b 1 is the slope. Statistical software will ...Missing: Cov( | Show results with:Cov(
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[PDF] Simple Linear Regression Models, with Hints at Their Estimation
In lecture 1, we saw that the optimal linear predictor of Y from X has slope β1 = Cov [X, Y ] /Var [X], and intercept β0 = E [Y ]-β1E [X]. A common tactic.