1.3: Shifting and Reflecting

Last updated

Oct 6, 2021
Save as PDF
- 1.2: Graphs and Symmetry
- 1.4: Composition and Inverses

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

1. Six Basic Functions

Below are six basic functions:

$SHIFT.HTM_txt_absx.gif$
$xsquare.gif$
$xCube.gif$
$SHIFT.HTM_txt_oneOverX.gif$
$rootx.gif$
$cubeRoot.gif$

Memorize the shapes of these functions.

2. Horizontal Shifting

Consider the graphs

$SHIFT.HTM_txt_RtShift.gif$

$y =$

$(x+0)^2$
$(x+1)^2$
$(x+2)^2$
$(x+3)^2$

Exercise

Use the list features of a calculator to sketch the graph of

$y = \dfrac{1}{ [x - \{0,1,2,3\}] }$

Horizontal Shifting Rules

Rule 1: $f(x - a) = f(x)$ shifted $a$ units to the right.
Rule 2: $f(x + a) = f(x)$ shifted $a$ units to the left.

3. Vertical Shifting

Consider the graphs

$SHIFT.HTM_txt_vertShft.gif$

$y =$

$x^3$
$x^3+ 1$
$x^3 + 2$
$x^3 + 3$

Exercise

Use the list features of a calculator to sketch the graph of

$y = x^3 - \{0,1,2,3\}$

Vertical Shifting Rules

Rule 3: $f(x ) + a = f(x)$ shifted a units up.
Rule 4: $f(x) - a = f(x)$ shifted a units down.

4. Reflecting About the x-axis

Consider the graphs of

$y = x^2$ and $y = -x^2$ .

$SHIFT.HTM_txt_xReflect.gif$

x-Axis Reflection Rule

Rule 5: $-f(x) = f(x)$ reflected about the x-axis.

5. Reflecting About the y-axis

Exercise

Use the calculator to graph

$y=\sqrt{x}$

and
$y=\sqrt{-x}$

y-Axis Reflection Rule

Rule 6: $f(-x ) = f(x)$ reflected about the y-axis.

6. Stretching and Compressing

Exercise

Graph the following:

$y = \{1,2,3,4\}x^3$

$y = {1/2,1/3,1/4,1/5}x^3$

Stretching and Compression rules:

Rule 7: $cf(x ) = f(x)$ (for $c > 1$ ) stretched vertically.
Rule 8: $cf(x ) = f(x)$ (for $c < 1$ ) compressed vertically.

Exercise

Graph the following

$y = x^2 - 10$
$y = \sqrt{x - 2}$
$y = -|x - 5| + 3$

We will do some examples (including the graph of the winnings for the gambler and for the casino).

7. Increasing and Decreasing Functions

Definition

A function is called increasing if as an object moves from left to right, it is moving upwards along the graph. Or equivalently,

If $x < y$ , then $f(x) < f(y)$ .

$SHIFT.HTM_txt_incDec.gif$

Example 1

The curve

$y = x^2$

is increasing on $(0,\infty)$ and decreasing on $(-\infty,0)$ .

Contributors

Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.

Search

Text Color

Text Size

Margin Size

Font Type

1. Six Basic Functions

2. Horizontal Shifting

3. Vertical Shifting

4. Reflecting About the x-axis

5. Reflecting About the y-axis

6. Stretching and Compressing

7. Increasing and Decreasing Functions

Contributors

Support Center

How can we help?