11: Linear Functions
- Page ID
- 174231
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 11.1: The Slope of a Line
- This page explains the concept of slope, detailing its measurement of a line's steepness as the ratio of vertical change to horizontal change. It defines essential terms like rate of change and average rate of change, presents the slope formula applicable to any two points, and categorizes slopes of horizontal and vertical lines.
- 11.2: Equations of Lines - Slope-Intercept Form
- This page covers the slope-intercept form of linear equations, \(y = mx + b\), explaining the slope \(m\) and \(y\)-intercept \(b\). It includes methods for converting equations into this form and characteristics of vertical and horizontal lines. Theorems highlight the unique representation of nonvertical lines. Additionally, it provides examples to identify slopes and \(y\)-intercepts from various equations, distinguishing different line types.
- 11.3: Equations of Lines - Point-Slope Form
- This page introduces the point-slope form of a line, \(y - y_1 = m(x - x_1)\), highlighting that vertical lines lack this form due to undefined slope. It includes theorems and examples to demonstrate how to use this form to derive equations from given slopes and points, convert them into slope-intercept and standard forms, and apply them in creating linear models for real-world scenarios.
- 11.4: Graphing Linear Equations
- This page explains the basics of graphing linear equations, including definitions, theorems, and the characteristics of horizontal and vertical lines. It covers linear equations, slope-intercept form, the concepts of slope and intercepts, and the relationship between points to define a line. Practical tips for verifying graph accuracy, examples of identifying slopes, rewriting equations, and graphing using intercepts are provided.
- 11.5: Linear Models
- This page covers linear functions and models, detailing their form \(f(x)=mx+b\), including definitions of slope and intercept. It illustrates how linear models represent real-world situations with constant rates of change and provides examples in various contexts such as costs and distance. The limitations of linear models in making predictions beyond reasonable ranges are also highlighted.
- 11.6: The Average Rate of Change of a Function
- This page explains the average rate of change in both linear and nonlinear functions, highlighting that linear functions maintain a constant average rate, equivalent to their slope, while nonlinear functions exhibit varying rates depending on chosen intervals. It includes definitions and examples for clearer understanding and underscores the significance of consistent subtraction in calculations.
- 11.7: Parallel and Perpendicular Lines
- This page covers definitions and theorems on parallel and perpendicular lines, highlighting that parallel lines share equal slopes and do not intersect, while perpendicular lines meet at right angles with slopes that are negative reciprocals.
- 11.8: Graphing Linear Inequalities in Two Variables
- This page explains linear inequalities in two variables, covering solution sets, boundary lines, and half-planes. It differentiates between strict and non-strict inequalities and discusses their graphical representation. The solution set is a half-plane determined by the boundary line, shaded based on inequality type. Examples illustrate using test points to identify the correct half-plane for different inequalities, including those with boundaries through the origin.