3.1E: Transformations of Functions (Exercises)

Last updated
Save as PDF

Page ID: 217527

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

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

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \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}} } $

$\newcommand{\longvect}{\overrightarrow}$

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

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Section 3.1 Exercises

Part 1: Vertical Stretches, Reflections, and Vertical Shifts

1. Write a function whose graph is the graph of $y=|x| $

Shifted up three units.
Vertically stretched by a factor of 3.
Vertically compressed by a factor of $\dfrac{1}{3} $, reflected about the x-axis, and then shifted down two units.

2. Write a function whose graph is the graph of $y=x^3$

Shifted down two units.
Vertically stretched by a factor of 4.
Vertically compressed by a factor of $\dfrac{1}{4} $ and reflected about the x-axis.

3. Use the graph of $y=f(x)$ below to sketch the graph of the indicated function with transformations. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$A piecewise defined function that decreases until x = -1, then is constant.$

$y=f(x)+2$
$y=2f(x)$
$y=-f(x)-1$
$y=-\dfrac{1}{2}f(x)+4$

4. Use the graph of $y=g(x)$ below to sketch the graph of the indicated function with transformations. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$the graph of a curve that increases from -4 to -2.3, then decreases until 2.3, and then increases again.$

$y=g(x)-3$
$y=\dfrac{1}{2}g(x)$
$y=-g(x)+1$
$y=-2g(x)+3$

5. Use the graph of $y=h(x)$ below to sketch the graph of the indicated function with transformations. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$a graph of a piecewise defined function that increase to y=3, is constant, then decreases to y = 1 at x =5.$

$y=h(x)-1$
$y=3h(x)$
$y=-h(x)+5$
$y=-\dfrac{1}{2}h(x)-2$

6. Use transformations of the graph of $y=x^2$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of the parabola y=x^2$

$y=x^2-2$
$y=3x^2$
$y=2x^2+1$
$y=-\dfrac{1}{2}x^2-3$

7. Use transformations of the graph of $y=x^3$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of the cubic function y = x^3$

$y=x^3+4$
$y=4x^3$
$y=\dfrac{1}{2}x^3-4 $
$y=-2\dfrac{1}{4}x^3+2 $

8. Use transformations of the graph of $y=\sqrt{x}$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of the square root function$

$y=\sqrt{x}-1$
$y=3\sqrt{x}$
$y=2\sqrt{x}+4$
$y=-\dfrac{1}{2}\sqrt{x}+2$

9. Use transformations of the graph of $y=\dfrac{1}{x+2}$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of a rational function with vertical asymptote at x=-2 and horizontal asymptote at y = 0.$

$y=\dfrac{1}{x+2}+2$
$y=2\dfrac{1}{x+2}$
$y=-\dfrac{1}{2}\dfrac{1}{x+2}-1$

10. Find a formula for the function $y=f(x)$ whose graph is given below.

$The graph of a downward facing parabola.$

11. Find a formula for the function $y=g(x)$ whose graph is given below.

$clipboard_ef343445c03b1c8c804be4b1fb6001d83.png$

12. Find a formula for the function $y=h(x)$ whose graph is given below.

$The graph of a shifted absolute value function.$

Past 2: Horizontal Shifts

13. Write a function whose graph is the graph of $y=|x| $

Shifted left three units.
Shifted right two units, then vertically stretched by a factor of 2.

14. Write a function whose graph is the graph of $y=x^3$

Shifted right one unit.
Shifted left five units, reflected about the y-axis, and shifted up one unit.

15. Use the graph of $y=f(x)$ below to sketch the graph of the indicated function with transformations. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$A piecewise defined function that decreases until x = -1, then is constant.$

$y=f(x+2)$
$y=3f(x-1)$
$y=-f(x+3)+4$

16. Use the graph of $y=g(x)$ below to sketch the graph of the indicated function with transformations. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$the graph of a curve that increases from -4 to -2.3, then decreases until 2.3, and then increases again.$

$y=g(x-2)$
$y=2g(x+3)$
$y=-g(x-1)+1$

17. Use the graph of $y=h(x)$ below to sketch the graph of the indicated function with transformations. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$a graph of a piecewise defined function that increase to y=3, is constant, then decreases to y = 1 at x =5.$

$y=h(x+1)$
$y=h(x-3)-3$
$y=-2h(x+2)$

18. Use transformations of the graph of $y=x^2$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of the parabola y=x^2$

$y=(x+4)^2-2$
$y=3(x-1)^2$

19. Use transformations of the graph of $y=x^3$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of the cubic function y = x^3$

$y=(x+3)^3$
$y=-(x-1)x^3+2$
$y=\dfrac{1}{2}(x+1)^3-4$

20. Use transformations of the graph of $y=\sqrt{x}$ below to sketch the graph of the indicated functions. Show a graph of each transformation if multiple transformations occur and where each indicated point was transformed.

$The graph of the square root function$

$y=\sqrt{x-4}$)
$y=\sqrt{x+2}-3$
$y=-2\sqrt{x-1}$

21. Find a formula for the function $y=f(x)$ whose graph is given below.

$The graph of a parabola$

22. Find a formula for the function $y=g(x)$ whose graph is given below.

$The graph of a reflected cubic.$

23. Find a formula for the function $y=h(x)$ whose graph is given below.

$clipboard_ef301e25162308678eee2ab06b0c8a987.png$

24. A manufacturer used test market data to predict the Revenue $R$ from producing $x$ units of one of their products with the function $R(x)=-1.12x^2+200x $.

If a new regulatory fee cause the revenue to decrease by $500, find a formula for the new revenue with this regulation.
If a new tariff caused the revenue to decrease 15%, find a formula for the new revenue with this regulation.

25. The amount of a therapeutic medication prescribed for a condition (in mg) is a function of the weight $w$ in pounds described by the function $R(x)=2.5\sqrt{x}+10$. If the scale weighting the patient overestimated the weight by 2.3 pounds, find a formula for the actual amount of the medication that should be prescribed.

Answer

1. a) $y=|x|+3$ b) $y=3|x|$ c) $y=-\dfrac{1}{3}|x|-2$

3. a) $the graph shifted up two units$