Question 151077


<a name="top">Table of Contents:



<a href="#table">Jump to Table</a>

<a href="#graph">Jump to Graph</a>



In order to graph {{{y=x^2-2x}}}, we need to plot some points.



Let's find y when {{{x=-3}}}

(Note: you can start with any x-value):



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(-3)^2-2(-3)}}} Plug in {{{x=-3}}}.



{{{y=1(9)-2(-3)}}} Square {{{-3}}} to get {{{9}}}.



{{{y=9-2(-3)}}} Multiply {{{1}}} and {{{9}}} to get {{{9}}}.



{{{y=9+6}}} Multiply {{{-2}}} and {{{-3}}} to get {{{6}}}.



{{{y=15}}} Combine like terms.



So when {{{x=-3}}}, then {{{y=15}}}.



So we have the point *[Tex \LARGE \left(-3,15\right)].



----------------------------------------------------



Let's find y when {{{x=-2}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(-2)^2-2(-2)}}} Plug in {{{x=-2}}}.



{{{y=1(4)-2(-2)}}} Square {{{-2}}} to get {{{4}}}.



{{{y=4-2(-2)}}} Multiply {{{1}}} and {{{4}}} to get {{{4}}}.



{{{y=4+4}}} Multiply {{{-2}}} and {{{-2}}} to get {{{4}}}.



{{{y=8}}} Combine like terms.



So when {{{x=-2}}}, then {{{y=8}}}.



So we have the point *[Tex \LARGE \left(-2,8\right)].



----------------------------------------------------



Let's find y when {{{x=-1}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(-1)^2-2(-1)}}} Plug in {{{x=-1}}}.



{{{y=1(1)-2(-1)}}} Square {{{-1}}} to get {{{1}}}.



{{{y=1-2(-1)}}} Multiply {{{1}}} and {{{1}}} to get {{{1}}}.



{{{y=1+2}}} Multiply {{{-2}}} and {{{-1}}} to get {{{2}}}.



{{{y=3}}} Combine like terms.



So when {{{x=-1}}}, then {{{y=3}}}.



So we have the point *[Tex \LARGE \left(-1,3\right)].



----------------------------------------------------



Let's find y when {{{x=0}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(0)^2-2(0)}}} Plug in {{{x=0}}}.



{{{y=1(0)-2(0)}}} Square {{{0}}} to get {{{0}}}.



{{{y=0-2(0)}}} Multiply {{{1}}} and {{{0}}} to get {{{0}}}.



{{{y=0+0}}} Multiply {{{-2}}} and {{{0}}} to get {{{0}}}.



{{{y=0}}} Combine like terms.



So when {{{x=0}}}, then {{{y=0}}}.



So we have the point *[Tex \LARGE \left(0,0\right)].



----------------------------------------------------



Let's find y when {{{x=1}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(1)^2-2(1)}}} Plug in {{{x=1}}}.



{{{y=1(1)-2(1)}}} Square {{{1}}} to get {{{1}}}.



{{{y=1-2(1)}}} Multiply {{{1}}} and {{{1}}} to get {{{1}}}.



{{{y=1-2}}} Multiply {{{-2}}} and {{{1}}} to get {{{-2}}}.



{{{y=-1}}} Combine like terms.



So when {{{x=1}}}, then {{{y=-1}}}.



So we have the point *[Tex \LARGE \left(1,-1\right)].



----------------------------------------------------



Let's find y when {{{x=2}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(2)^2-2(2)}}} Plug in {{{x=2}}}.



{{{y=1(4)-2(2)}}} Square {{{2}}} to get {{{4}}}.



{{{y=4-2(2)}}} Multiply {{{1}}} and {{{4}}} to get {{{4}}}.



{{{y=4-4}}} Multiply {{{-2}}} and {{{2}}} to get {{{-4}}}.



{{{y=0}}} Combine like terms.



So when {{{x=2}}}, then {{{y=0}}}.



So we have the point *[Tex \LARGE \left(2,0\right)].



----------------------------------------------------



Let's find y when {{{x=3}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(3)^2-2(3)}}} Plug in {{{x=3}}}.



{{{y=1(9)-2(3)}}} Square {{{3}}} to get {{{9}}}.



{{{y=9-2(3)}}} Multiply {{{1}}} and {{{9}}} to get {{{9}}}.



{{{y=9-6}}} Multiply {{{-2}}} and {{{3}}} to get {{{-6}}}.



{{{y=3}}} Combine like terms.



So when {{{x=3}}}, then {{{y=3}}}.



So we have the point *[Tex \LARGE \left(3,3\right)].



----------------------------------------------------



Let's find y when {{{x=4}}}:



{{{y=x^2-2x}}} Start with the given equation.



{{{y=(4)^2-2(4)}}} Plug in {{{x=4}}}.



{{{y=1(16)-2(4)}}} Square {{{4}}} to get {{{16}}}.



{{{y=16-2(4)}}} Multiply {{{1}}} and {{{16}}} to get {{{16}}}.



{{{y=16-8}}} Multiply {{{-2}}} and {{{4}}} to get {{{-8}}}.



{{{y=8}}} Combine like terms.



So when {{{x=4}}}, then {{{y=8}}}.



So we have the point *[Tex \LARGE \left(4,8\right)].



----------------------------------------------------



<a name="table">



<a href="#top">Jump to Top of Page</a>

Now let's make a table of the values we just found.



<h4>Table of Values:</h4><pre>

<TABLE border="1" width="100">
<TR><TD>x</TD><TD>y</TD></TR><tr><td>-3</td><td>15</td></tr>
<tr><td>-2</td><td>8</td></tr>
<tr><td>-1</td><td>3</td></tr>
<tr><td>0</td><td>0</td></tr>
<tr><td>1</td><td>-1</td></tr>
<tr><td>2</td><td>0</td></tr>
<tr><td>3</td><td>3</td></tr>
<tr><td>4</td><td>8</td></tr>
</TABLE>

</pre>

Now let's plot the points:



{{{ drawing(900, 900, -10, 10, -10, 10,
grid(1),
graph(900, 900, -10, 10, -10, 10, 0),
circle(-3,15,0.08),circle(-3,15,0.10),
circle(-2,8,0.08),circle(-2,8,0.10),
circle(-1,3,0.08),circle(-1,3,0.10),
circle(0,0,0.08),circle(0,0,0.10),
circle(1,-1,0.08),circle(1,-1,0.10),
circle(2,0,0.08),circle(2,0,0.10),
circle(3,3,0.08),circle(3,3,0.10),
circle(4,8,0.08),circle(4,8,0.10)

)}}}


<a name="graph">



<a href="#top">Jump to Top of Page</a>

<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=x^2-2x}}}:



{{{ drawing(900, 900, -10, 10, -10, 10,
grid(1),
graph(900, 900, -10, 10, -10, 10, x^2-2x),
circle(-3,15,0.08),circle(-3,15,0.10),
circle(-2,8,0.08),circle(-2,8,0.10),
circle(-1,3,0.08),circle(-1,3,0.10),
circle(0,0,0.08),circle(0,0,0.10),
circle(1,-1,0.08),circle(1,-1,0.10),
circle(2,0,0.08),circle(2,0,0.10),
circle(3,3,0.08),circle(3,3,0.10),
circle(4,8,0.08),circle(4,8,0.10)

)}}} Graph of {{{y=x^2-2x}}}