Question 148855


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In order to graph {{{y=x^3+4x^2+x-6}}}, we need to plot some points.



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

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



{{{y=x^3+4x^2+x-6}}} Start with the given equation.



{{{y=(-3)^3+4(-3)^2+-3-6}}} Plug in {{{x=-3}}}.



{{{y=1(-27)+4(-3)^2-3-6}}} Cube {{{-3}}} to get {{{-27}}}.



{{{y=1(-27)+4(9)-3-6}}} Square {{{-3}}} to get {{{9}}}.



{{{y=-27+4(9)-3-6}}} Multiply {{{1}}} and {{{-27}}} to get {{{-27}}}.



{{{y=-27+36-3-6}}} Multiply {{{4}}} and {{{9}}} to get {{{36}}}.



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



So when {{{x=-3}}}, then {{{y=0}}}.



So we have the point *[Tex \LARGE \left(-3,0\right)].



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



{{{y=x^3+4x^2+x-6}}} Start with the given equation.



{{{y=(-2)^3+4(-2)^2+-2-6}}} Plug in {{{x=-2}}}.



{{{y=1(-8)+4(-2)^2-2-6}}} Cube {{{-2}}} to get {{{-8}}}.



{{{y=1(-8)+4(4)-2-6}}} Square {{{-2}}} to get {{{4}}}.



{{{y=-8+4(4)-2-6}}} Multiply {{{1}}} and {{{-8}}} to get {{{-8}}}.



{{{y=-8+16-2-6}}} Multiply {{{4}}} and {{{4}}} to get {{{16}}}.



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



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



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



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



{{{y=x^3+4x^2+x-6}}} Start with the given equation.



{{{y=(-1)^3+4(-1)^2+-1-6}}} Plug in {{{x=-1}}}.



{{{y=1(-1)+4(-1)^2-1-6}}} Cube {{{-1}}} to get {{{-1}}}.



{{{y=1(-1)+4(1)-1-6}}} Square {{{-1}}} to get {{{1}}}.



{{{y=-1+4(1)-1-6}}} Multiply {{{1}}} and {{{-1}}} to get {{{-1}}}.



{{{y=-1+4-1-6}}} Multiply {{{4}}} and {{{1}}} to get {{{4}}}.



{{{y=-4}}} Combine like terms.



So when {{{x=-1}}}, then {{{y=-4}}}.



So we have the point *[Tex \LARGE \left(-1,-4\right)].



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



{{{y=x^3+4x^2+x-6}}} Start with the given equation.



{{{y=(0)^3+4(0)^2+0-6}}} Plug in {{{x=0}}}.



{{{y=1(0)+4(0)^2+0-6}}} Cube {{{0}}} to get {{{0}}}.



{{{y=1(0)+4(0)+0-6}}} Square {{{0}}} to get {{{0}}}.



{{{y=0+4(0)+0-6}}} Multiply {{{1}}} and {{{0}}} to get {{{0}}}.



{{{y=0+0+0-6}}} Multiply {{{4}}} and {{{0}}} to get {{{0}}}.



{{{y=-6}}} Combine like terms.



So when {{{x=0}}}, then {{{y=-6}}}.



So we have the point *[Tex \LARGE \left(0,-6\right)].



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



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



{{{y=x^3+4x^2+x-6}}} Start with the given equation.



{{{y=(1)^3+4(1)^2+1-6}}} Plug in {{{x=1}}}.



{{{y=1(1)+4(1)^2+1-6}}} Cube {{{1}}} to get {{{1}}}.



{{{y=1(1)+4(1)+1-6}}} Square {{{1}}} to get {{{1}}}.



{{{y=1+4(1)+1-6}}} Multiply {{{1}}} and {{{1}}} to get {{{1}}}.



{{{y=1+4+1-6}}} Multiply {{{4}}} and {{{1}}} to get {{{4}}}.



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



So when {{{x=1}}}, then {{{y=0}}}.



So we have the point *[Tex \LARGE \left(1,0\right)].



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



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



{{{y=x^3+4x^2+x-6}}} Start with the given equation.



{{{y=(2)^3+4(2)^2+2-6}}} Plug in {{{x=2}}}.



{{{y=1(8)+4(2)^2+2-6}}} Cube {{{2}}} to get {{{8}}}.



{{{y=1(8)+4(4)+2-6}}} Square {{{2}}} to get {{{4}}}.



{{{y=8+4(4)+2-6}}} Multiply {{{1}}} and {{{8}}} to get {{{8}}}.



{{{y=8+16+2-6}}} Multiply {{{4}}} and {{{4}}} to get {{{16}}}.



{{{y=20}}} Combine like terms.



So when {{{x=2}}}, then {{{y=20}}}.



So we have the point *[Tex \LARGE \left(2,20\right)].



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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>0</td></tr>
<tr><td>-2</td><td>0</td></tr>
<tr><td>-1</td><td>-4</td></tr>
<tr><td>0</td><td>-6</td></tr>
<tr><td>1</td><td>0</td></tr>
<tr><td>2</td><td>20</td></tr>
</TABLE>

</pre>

Now let's plot the points:



{{{ drawing(900, 900, -15, 15, -15, 22,
grid(1),
graph(900, 900, -15, 15, -15, 22, 0),
circle(-3,0,0.08),circle(-3,0,0.10),
circle(-2,0,0.08),circle(-2,0,0.10),
circle(-1,-4,0.08),circle(-1,-4,0.10),
circle(0,-6,0.08),circle(0,-6,0.10),
circle(1,0,0.08),circle(1,0,0.10),
circle(2,20,0.08),circle(2,20,0.10)

)}}}


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<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=x^3+4x^2+x-6}}}:



{{{ drawing(900, 900, -15, 15, -15, 22,
grid(1),
graph(900, 900, -15, 15, -15, 22, x^3+4x^2+x-6),
circle(-3,0,0.08),circle(-3,0,0.10),
circle(-2,0,0.08),circle(-2,0,0.10),
circle(-1,-4,0.08),circle(-1,-4,0.10),
circle(0,-6,0.08),circle(0,-6,0.10),
circle(1,0,0.08),circle(1,0,0.10),
circle(2,20,0.08),circle(2,20,0.10)

)}}} Graph of {{{y=x^3+4x^2+x-6}}}