4.5: Toolbox Functions
- Page ID
- 40918
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning the general shapes of some common function families can be helpful in analyzing various problems. This can also be helpful in applying the ideas of statistical regression. Statistical regression typically gathers a collection of data points and tries to fit a mathematical function to the data points. Choosing the type of function that will best fit the data is an important step in determining a suitable regression function.
Below is an example of a Linear Regression:
The graph below is an example of a Quadratic Regression:
Being familiar with the typical shape of the various function families can help in analyzing experimental data.
The standard function families are:
Some other function families that we won't discuss are:
Exponential function: \(\quad f(x)=a^{x}\)
Logarithmic function: \(f(x)=\log _{b} x\)
Trigonometric function: \(\quad f(x)=\sin x\)
Exercises 4.5
Sketch the graph for each of the following transformations.
1) \(\quad f(x)=x^{2}+3\)
2) \(\quad f(x)=x^{2}-4\)
3) \(\quad f(x)=(x-5)^{2}+3\)
4) \(\quad f(x)=(x+1)^{2}-4\)
5) \(\quad f(x)=|x|-2\)
6) \(\quad f(x)=|x|+5\)
7) \(\quad f(x)=|x+3|-2\)
8) \(\quad f(x)=|x-1|+5\)
Match each of the following equations to the appropriate graph.
9) \(\quad f(x)=\sqrt{x+4}-1\)
10) \(\quad h(x)=x^{2}+x-6\)
11) \(\quad g(x)=|x+2|-3\)
12) \(\quad f(x)=4 x-x^{2}\)
13) \(\quad h(x)=-|x-2|+1\)
14) \(\quad g(x)=-\frac{3}{5} x+2\)
15) \(\quad h(x)=\sqrt[3]{x}+4\)
16) \(\quad f(x)=x^{3}-3 x^{2}+3 x+2\)
17) \(\quad g(x)=-\sqrt[3]{x+2}\)
18) \(\quad h(x)=-\sqrt{x+1}-3\)
19) \(\quad f(x)=\frac{5}{4} x-2\)
20) \(\quad f(x)=4 x-x^{3}\)
