<PRE><B>Question 150910</B>
# 1






Let&#39;s find the function value when {{{x=1}}}:



{{{y=x^2-3x+2}}} Start with the given equation.



{{{y=(1)^2-3(1)+2}}} Plug in {{{x=1}}}.



{{{y=1(1)-3(1)+2}}} Square {{{1}}} to get {{{1}}}.



{{{y=1-3(1)+2}}} Multiply {{{1}}} and {{{1}}} to get {{{1}}}.



{{{y=1-3+2}}} Multiply {{{-3}}} and {{{1}}} to get {{{-3}}}.



{{{y=0}}} Combine like terms.



So if {{{x=1}}}, then {{{y=0}}}.


-------------


Let&#39;s find the function value when {{{x=2}}}:



{{{y=x^2-3x+2}}} Start with the given equation.



{{{y=(2)^2-3(2)+2}}} Plug in {{{x=2}}}.



{{{y=1(4)-3(2)+2}}} Square {{{2}}} to get {{{4}}}.



{{{y=4-3(2)+2}}} Multiply {{{1}}} and {{{4}}} to get {{{4}}}.



{{{y=4-6+2}}} Multiply {{{-3}}} and {{{2}}} to get {{{-6}}}.



{{{y=0}}} Combine like terms.



So if {{{x=2}}}, then {{{y=0}}}.


-------------


Let&#39;s find the function value when {{{x=4}}}:



{{{y=x^2-3x+2}}} Start with the given equation.



{{{y=(4)^2-3(4)+2}}} Plug in {{{x=4}}}.



{{{y=1(16)-3(4)+2}}} Square {{{4}}} to get {{{16}}}.



{{{y=16-3(4)+2}}} Multiply {{{1}}} and {{{16}}} to get {{{16}}}.



{{{y=16-12+2}}} Multiply {{{-3}}} and {{{4}}} to get {{{-12}}}.



{{{y=6}}} Combine like terms.



So if {{{x=4}}}, then {{{y=6}}}.


-------------


Let&#39;s find the function value when {{{x=8}}}:



{{{y=x^2-3x+2}}} Start with the given equation.



{{{y=(8)^2-3(8)+2}}} Plug in {{{x=8}}}.



{{{y=1(64)-3(8)+2}}} Square {{{8}}} to get {{{64}}}.



{{{y=64-3(8)+2}}} Multiply {{{1}}} and {{{64}}} to get {{{64}}}.



{{{y=64-24+2}}} Multiply {{{-3}}} and {{{8}}} to get {{{-24}}}.



{{{y=42}}} Combine like terms.



So if {{{x=8}}}, then {{{y=42}}}.


-------------


Let&#39;s find the function value when {{{x=16}}}:



{{{y=x^2-3x+2}}} Start with the given equation.



{{{y=(16)^2-3(16)+2}}} Plug in {{{x=16}}}.



{{{y=1(256)-3(16)+2}}} Square {{{16}}} to get {{{256}}}.



{{{y=256-3(16)+2}}} Multiply {{{1}}} and {{{256}}} to get {{{256}}}.



{{{y=256-48+2}}} Multiply {{{-3}}} and {{{16}}} to get {{{-48}}}.



{{{y=210}}} Combine like terms.



So if {{{x=16}}}, then {{{y=210}}}.


-------------


Now let&#39;s make a table of the values we just found.



&lt;a name=&quot;table&quot;&gt;



&lt;a href=&quot;#top&quot;&gt;Jump to Top of Page&lt;/a&gt;

&lt;h4&gt;Table of Values:&lt;/h4&gt;&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;6&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;42&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;210&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;




&lt;hr&gt;


# 2





Let&#39;s find the function value when {{{x=1}}}:



{{{y=2x^3+7x^2-x-1}}} Start with the given equation.



{{{y=2(1)^3+7(1)^2-1-1}}} Plug in {{{x=1}}}.



{{{y=2(1)+7(1)^2-1-1}}} Cube {{{1}}} to get {{{1}}}.



{{{y=2(1)+7(1)-1-1}}} Square {{{1}}} to get {{{1}}}.



{{{y=2+7(1)-1-1}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}.



{{{y=2+7-1-1}}} Multiply {{{7}}} and {{{1}}} to get {{{7}}}.



{{{y=7}}} Combine like terms.



So if {{{x=1}}}, then {{{y=7}}}.


-------------


Let&#39;s find the function value when {{{x=2}}}:



{{{y=2x^3+7x^2-x-1}}} Start with the given equation.



{{{y=2(2)^3+7(2)^2-2-1}}} Plug in {{{x=2}}}.



{{{y=2(8)+7(2)^2-2-1}}} Cube {{{2}}} to get {{{8}}}.



{{{y=2(8)+7(4)-2-1}}} Square {{{2}}} to get {{{4}}}.



{{{y=16+7(4)-2-1}}} Multiply {{{2}}} and {{{8}}} to get {{{16}}}.



{{{y=16+28-2-1}}} Multiply {{{7}}} and {{{4}}} to get {{{28}}}.



{{{y=41}}} Combine like terms.



So if {{{x=2}}}, then {{{y=41}}}.


-------------


Let&#39;s find the function value when {{{x=4}}}:



{{{y=2x^3+7x^2-x-1}}} Start with the given equation.



{{{y=2(4)^3+7(4)^2-4-1}}} Plug in {{{x=4}}}.



{{{y=2(64)+7(4)^2-4-1}}} Cube {{{4}}} to get {{{64}}}.



{{{y=2(64)+7(16)-4-1}}} Square {{{4}}} to get {{{16}}}.



{{{y=128+7(16)-4-1}}} Multiply {{{2}}} and {{{64}}} to get {{{128}}}.



{{{y=128+112-4-1}}} Multiply {{{7}}} and {{{16}}} to get {{{112}}}.



{{{y=235}}} Combine like terms.



So if {{{x=4}}}, then {{{y=235}}}.


-------------


Let&#39;s find the function value when {{{x=8}}}:



{{{y=2x^3+7x^2-x-1}}} Start with the given equation.



{{{y=2(8)^3+7(8)^2-8-1}}} Plug in {{{x=8}}}.



{{{y=2(512)+7(8)^2-8-1}}} Cube {{{8}}} to get {{{512}}}.



{{{y=2(512)+7(64)-8-1}}} Square {{{8}}} to get {{{64}}}.



{{{y=1024+7(64)-8-1}}} Multiply {{{2}}} and {{{512}}} to get {{{1024}}}.



{{{y=1024+448-8-1}}} Multiply {{{7}}} and {{{64}}} to get {{{448}}}.



{{{y=1463}}} Combine like terms.



So if {{{x=8}}}, then {{{y=1463}}}.


-------------


Let&#39;s find the function value when {{{x=16}}}:



{{{y=2x^3+7x^2-x-1}}} Start with the given equation.



{{{y=2(16)^3+7(16)^2-16-1}}} Plug in {{{x=16}}}.



{{{y=2(4096)+7(16)^2-16-1}}} Cube {{{16}}} to get {{{4096}}}.



{{{y=2(4096)+7(256)-16-1}}} Square {{{16}}} to get {{{256}}}.



{{{y=8192+7(256)-16-1}}} Multiply {{{2}}} and {{{4096}}} to get {{{8192}}}.



{{{y=8192+1792-16-1}}} Multiply {{{7}}} and {{{256}}} to get {{{1792}}}.



{{{y=9967}}} Combine like terms.



So if {{{x=16}}}, then {{{y=9967}}}.


-------------


Now let&#39;s make a table of the values we just found.




&lt;h4&gt;Table of Values:&lt;/h4&gt;&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;7&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;41&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;235&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;1463&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;9967&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;



&lt;hr&gt;



# 3





Let&#39;s find the function value when {{{x=1}}}:



{{{y=10x}}} Start with the given equation.



{{{y=10(1)}}} Plug in {{{x=1}}}.



{{{y=10}}} Multiply {{{10}}} and {{{1}}} to get {{{10}}}.



{{{y=10}}} Combine like terms.



So if {{{x=1}}}, then {{{y=10}}}.


-------------


Let&#39;s find the function value when {{{x=2}}}:



{{{y=10x}}} Start with the given equation.



{{{y=10(2)}}} Plug in {{{x=2}}}.



{{{y=20}}} Multiply {{{10}}} and {{{2}}} to get {{{20}}}.



{{{y=20}}} Combine like terms.



So if {{{x=2}}}, then {{{y=20}}}.


-------------


Let&#39;s find the function value when {{{x=4}}}:



{{{y=10x}}} Start with the given equation.



{{{y=10(4)}}} Plug in {{{x=4}}}.



{{{y=40}}} Multiply {{{10}}} and {{{4}}} to get {{{40}}}.



{{{y=40}}} Combine like terms.



So if {{{x=4}}}, then {{{y=40}}}.


-------------


Let&#39;s find the function value when {{{x=8}}}:



{{{y=10x}}} Start with the given equation.



{{{y=10(8)}}} Plug in {{{x=8}}}.



{{{y=80}}} Multiply {{{10}}} and {{{8}}} to get {{{80}}}.



{{{y=80}}} Combine like terms.



So if {{{x=8}}}, then {{{y=80}}}.


-------------


Let&#39;s find the function value when {{{x=16}}}:



{{{y=10x}}} Start with the given equation.



{{{y=10(16)}}} Plug in {{{x=16}}}.



{{{y=160}}} Multiply {{{10}}} and {{{16}}} to get {{{160}}}.



{{{y=160}}} Combine like terms.



So if {{{x=16}}}, then {{{y=160}}}.


-------------


Now let&#39;s make a table of the values we just found.



&lt;h4&gt;Table of Values:&lt;/h4&gt;&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;10&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;20&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;40&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;80&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;160&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;




&lt;hr&gt;



# 4


Note: ln(x) also looks like LN(x) (to pronounce it, simply read off the letters &quot;L&quot; &quot;N&quot;)


This is the natural log of x. So it is a logarithmic function.


Let&#39;s find the y value when {{{x=1}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(1)}}} Plug in {{{x=1}}}.



{{{y=0}}} Take the natural log of 1 to get 0



So when {{{x=1}}}, {{{y=0}}}. 



----------------------------



Let&#39;s find the y value when {{{x=2}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(2)}}} Plug in {{{x=2}}}.



{{{y=0.693}}} Take the natural log of 2 to get 0.693



So when {{{x=2}}}, {{{y=0.693}}}.




----------------------------



Let&#39;s find the y value when {{{x=4}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(4)}}} Plug in {{{x=4}}}.



{{{y=1.386}}} Take the natural log of 4 to get 1.386



So when {{{x=4}}}, {{{y=1.386}}}. 


-------------------------------


Let&#39;s find the y value when {{{x=8}}} 



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(8)}}} Plug in {{{x=8}}}.



{{{y=2.079}}} Take the natural log of 8 to get 2.079



So when {{{x=8}}}, {{{y=2.079}}}. 



-------------------------------------



Let&#39;s find the y value when {{{x=16}}}  



{{{y=ln(x)}}} Start with the given equation.



{{{y=ln(16)}}} Plug in {{{x=16}}}.



{{{y=2.773}}} Take the natural log of 16 to get 2.773



So when {{{x=16}}}, {{{y=2.773}}}.



Now let&#39;s make a table of the values we just found.




&lt;h4&gt;Table of Values:&lt;/h4&gt;&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;0.693&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;1.386&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;2.079&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;2.773&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;





===========================================================


So to compare the rates, simply compare the tables:


Table for {{{y=x^2-3x+2}}}

&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;6&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;42&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;210&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;



Table for {{{y=2x^3+7x^2-x-1}}}


&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;7&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;41&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;235&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;1463&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;9967&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;


Table for {{{y=10x}}}


&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;10&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;20&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;40&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;80&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;160&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;


Table for {{{y=ln(x)}}}


&lt;pre&gt;

&lt;TABLE border=&quot;1&quot; width=&quot;100&quot;&gt;
&lt;TR&gt;&lt;TD&gt;x&lt;/TD&gt;&lt;TD&gt;y&lt;/TD&gt;&lt;/TR&gt;&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;0.693&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;1.386&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;2.079&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;2.773&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;



From the tables, the functions can be sorted from slowest growing to fastest growing like this:


{{{y=ln(x)}}}, {{{y=10x}}}, {{{y=x^2-3x+2}}}, and {{{y=2x^3+7x^2-x-1}}}




So {{{y=ln(x)}}} is the slowest growing function and {{{y=2x^3+7x^2-x-1}}} is the fastest growing function.




If all of that is confusing, simply graph all of the functions to get


{{{ drawing(500, 500, -10, 10, -10, 20,
 graph( 500, 500, -10, 10, -10, 20,x^2-3x+2,2x^3+7x^2-x-1,10x,ln(x))

)}}} Graph of {{{y=x^2-3x+2}}} (red), {{{y=2x^3+7x^2-x-1}}} (green), {{{y=10x}}}(blue), and {{{y=ln(x)}}} (purple)


From the graph, we can see that the order from slowest growing to fastest growing is


{{{y=ln(x)}}}, {{{y=10x}}}, {{{y=x^2-3x+2}}}, and {{{y=2x^3+7x^2-x-1}}}
</PRE>