10: Graphing Functions
- Page ID
- 173508
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.1: Intercepts and Base Graphs
- This page covers the interpretation of graphs, explaining \(y\)- and \(x\)-intercepts, highlighting that a function may have one \(y\)-intercept and multiple \(x\)-intercepts. It introduces four main parent functions: squaring, cubing, absolute value, and square root, detailing their characteristics.
- 10.2: Symmetry
- This page defines even and odd functions, explaining their properties, symmetry, and classification methods. Even functions satisfy \(f(-x) = f(x)\) and are symmetric about the \(y\)-axis, whereas odd functions satisfy \(f(-x) = -f(x)\) and exhibit symmetry about the origin.
- 10.3: Intervals of Increase and Decrease
- This page presents key definitions and theorems regarding the behavior of functions, focusing on increasing, decreasing, and constant functions in relation to intervals and local extrema. It highlights the significance of open intervals at points of change and discusses the property of monotonic functions being one-to-one. Examples demonstrate the application of these concepts through graphical and algebraic methods, while also emphasizing the importance of considering the domain.
- 10.4: Local/Relative and Global/Absolute Extrema
- This page covers the concepts of extrema within functions, focusing on absolute and local maxima and minima. It clarifies the distinction between absolute (global) and local (relative) extrema, supported by the Extreme Value Theorem, which asserts that a continuous function on a closed interval reaches both absolute maximum and minimum values.
- 10.5: Transformations - Rigid Transformations
- This page covers the transformations of function graphs, introducing rigid transformations like translations (shifting) and reflections (mirror images), alongside their associated definitions and theorems. It elaborates on vertical and horizontal transformations, explaining how they modify functions based on constants. Key examples illustrate combinations of these transformations, highlighting the importance of systematic ordering to achieve new function graphs.
- 10.6: Transformations - Non-Rigid Transformations
- This page of the textbook covers non-rigid transformations of graphs, focusing on vertical and horizontal stretches or compressions influenced by constants \(a\) and \(b\). It highlights how these transformations alter the graph's size and shape, with specific attention to the counterintuitive nature of horizontal scaling.
- 10.7: Transformations - Combining Transformations
- This page covers combined transformations of functions using the equation \(g(x)=a\,f(b(x-h))+k\), detailing how parameters affect graph transformations like scaling, reflection, and shifting. It emphasizes the order of operations for accurate representation.
- 10.8: Piecewise Functions
- This page explains piecewise-defined functions, detailing their structure with multiple sub-functions for specific intervals. It emphasizes the need for clear definitions to avoid ambiguity at breakpoints and the importance of inequality symbols in evaluation. The page includes examples for evaluating and graphing piecewise functions, addressing jump discontinuities, and cites a Wikipedia source under a Creative Commons license, along with worksheets for further practice.