Question 153191


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In order to graph {{{y=-x^2-8x+9}}}, we need to plot some points.



Let's find y when {{{x=-9}}}

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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-9)^2-8(-9)+9}}} Plug in {{{x=-9}}}.



{{{y=-1(81)-8(-9)+9}}} Square {{{-9}}} to get {{{81}}}.



{{{y=-81-8(-9)+9}}} Multiply {{{-1}}} and {{{81}}} to get {{{-81}}}.



{{{y=-81+72+9}}} Multiply {{{-8}}} and {{{-9}}} to get {{{72}}}.



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



So when {{{x=-9}}}, then {{{y=0}}}.



So we have the point *[Tex \LARGE \left(-9,0\right)].



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



Let's find y when {{{x=-8}}}:



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-8)^2-8(-8)+9}}} Plug in {{{x=-8}}}.



{{{y=-1(64)-8(-8)+9}}} Square {{{-8}}} to get {{{64}}}.



{{{y=-64-8(-8)+9}}} Multiply {{{-1}}} and {{{64}}} to get {{{-64}}}.



{{{y=-64+64+9}}} Multiply {{{-8}}} and {{{-8}}} to get {{{64}}}.



{{{y=9}}} Combine like terms.



So when {{{x=-8}}}, then {{{y=9}}}.



So we have the point *[Tex \LARGE \left(-8,9\right)].



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



Let's find y when {{{x=-7}}}:



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-7)^2-8(-7)+9}}} Plug in {{{x=-7}}}.



{{{y=-1(49)-8(-7)+9}}} Square {{{-7}}} to get {{{49}}}.



{{{y=-49-8(-7)+9}}} Multiply {{{-1}}} and {{{49}}} to get {{{-49}}}.



{{{y=-49+56+9}}} Multiply {{{-8}}} and {{{-7}}} to get {{{56}}}.



{{{y=16}}} Combine like terms.



So when {{{x=-7}}}, then {{{y=16}}}.



So we have the point *[Tex \LARGE \left(-7,16\right)].



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



Let's find y when {{{x=-6}}}:



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-6)^2-8(-6)+9}}} Plug in {{{x=-6}}}.



{{{y=-1(36)-8(-6)+9}}} Square {{{-6}}} to get {{{36}}}.



{{{y=-36-8(-6)+9}}} Multiply {{{-1}}} and {{{36}}} to get {{{-36}}}.



{{{y=-36+48+9}}} Multiply {{{-8}}} and {{{-6}}} to get {{{48}}}.



{{{y=21}}} Combine like terms.



So when {{{x=-6}}}, then {{{y=21}}}.



So we have the point *[Tex \LARGE \left(-6,21\right)].



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



Let's find y when {{{x=-5}}}:



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-5)^2-8(-5)+9}}} Plug in {{{x=-5}}}.



{{{y=-1(25)-8(-5)+9}}} Square {{{-5}}} to get {{{25}}}.



{{{y=-25-8(-5)+9}}} Multiply {{{-1}}} and {{{25}}} to get {{{-25}}}.



{{{y=-25+40+9}}} Multiply {{{-8}}} and {{{-5}}} to get {{{40}}}.



{{{y=24}}} Combine like terms.



So when {{{x=-5}}}, then {{{y=24}}}.



So we have the point *[Tex \LARGE \left(-5,24\right)].



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



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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-4)^2-8(-4)+9}}} Plug in {{{x=-4}}}.



{{{y=-1(16)-8(-4)+9}}} Square {{{-4}}} to get {{{16}}}.



{{{y=-16-8(-4)+9}}} Multiply {{{-1}}} and {{{16}}} to get {{{-16}}}.



{{{y=-16+32+9}}} Multiply {{{-8}}} and {{{-4}}} to get {{{32}}}.



{{{y=25}}} Combine like terms.



So when {{{x=-4}}}, then {{{y=25}}}.



So we have the point *[Tex \LARGE \left(-4,25\right)].



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



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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-3)^2-8(-3)+9}}} Plug in {{{x=-3}}}.



{{{y=-1(9)-8(-3)+9}}} Square {{{-3}}} to get {{{9}}}.



{{{y=-9-8(-3)+9}}} Multiply {{{-1}}} and {{{9}}} to get {{{-9}}}.



{{{y=-9+24+9}}} Multiply {{{-8}}} and {{{-3}}} to get {{{24}}}.



{{{y=24}}} Combine like terms.



So when {{{x=-3}}}, then {{{y=24}}}.



So we have the point *[Tex \LARGE \left(-3,24\right)].



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



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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-2)^2-8(-2)+9}}} Plug in {{{x=-2}}}.



{{{y=-1(4)-8(-2)+9}}} Square {{{-2}}} to get {{{4}}}.



{{{y=-4-8(-2)+9}}} Multiply {{{-1}}} and {{{4}}} to get {{{-4}}}.



{{{y=-4+16+9}}} Multiply {{{-8}}} and {{{-2}}} to get {{{16}}}.



{{{y=21}}} Combine like terms.



So when {{{x=-2}}}, then {{{y=21}}}.



So we have the point *[Tex \LARGE \left(-2,21\right)].



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



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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(-1)^2-8(-1)+9}}} Plug in {{{x=-1}}}.



{{{y=-1(1)-8(-1)+9}}} Square {{{-1}}} to get {{{1}}}.



{{{y=-1-8(-1)+9}}} Multiply {{{-1}}} and {{{1}}} to get {{{-1}}}.



{{{y=-1+8+9}}} Multiply {{{-8}}} and {{{-1}}} to get {{{8}}}.



{{{y=16}}} Combine like terms.



So when {{{x=-1}}}, then {{{y=16}}}.



So we have the point *[Tex \LARGE \left(-1,16\right)].



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



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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(0)^2-8(0)+9}}} Plug in {{{x=0}}}.



{{{y=-1(0)-8(0)+9}}} Square {{{0}}} to get {{{0}}}.



{{{y=0-8(0)+9}}} Multiply {{{-1}}} and {{{0}}} to get {{{0}}}.



{{{y=0+0+9}}} Multiply {{{-8}}} and {{{0}}} to get {{{0}}}.



{{{y=9}}} Combine like terms.



So when {{{x=0}}}, then {{{y=9}}}.



So we have the point *[Tex \LARGE \left(0,9\right)].



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



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



{{{y=-x^2-8x+9}}} Start with the given equation.



{{{y=-(1)^2-8(1)+9}}} Plug in {{{x=1}}}.



{{{y=-1(1)-8(1)+9}}} Square {{{1}}} to get {{{1}}}.



{{{y=-1-8(1)+9}}} Multiply {{{-1}}} and {{{1}}} to get {{{-1}}}.



{{{y=-1-8+9}}} Multiply {{{-8}}} and {{{1}}} to get {{{-8}}}.



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



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



So we have the point *[Tex \LARGE \left(1,0\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>-9</td><td>0</td></tr>
<tr><td>-8</td><td>9</td></tr>
<tr><td>-7</td><td>16</td></tr>
<tr><td>-6</td><td>21</td></tr>
<tr><td>-5</td><td>24</td></tr>
<tr><td>-4</td><td>25</td></tr>
<tr><td>-3</td><td>24</td></tr>
<tr><td>-2</td><td>21</td></tr>
<tr><td>-1</td><td>16</td></tr>
<tr><td>0</td><td>9</td></tr>
<tr><td>1</td><td>0</td></tr>
</TABLE>

</pre>

Now let's plot the points:



{{{ drawing(900, 900, -10, 10, -5, 27,
grid(1),
graph(900, 900, -10, 10, -5, 27, 0),
circle(-9,0,0.08),circle(-9,0,0.10),
circle(-8,9,0.08),circle(-8,9,0.10),
circle(-7,16,0.08),circle(-7,16,0.10),
circle(-6,21,0.08),circle(-6,21,0.10),
circle(-5,24,0.08),circle(-5,24,0.10),
circle(-4,25,0.08),circle(-4,25,0.10),
circle(-3,24,0.08),circle(-3,24,0.10),
circle(-2,21,0.08),circle(-2,21,0.10),
circle(-1,16,0.08),circle(-1,16,0.10),
circle(0,9,0.08),circle(0,9,0.10),
circle(1,0,0.08),circle(1,0,0.10)

)}}}


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<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=-x^2-8x+9}}}:



{{{ drawing(900, 900, -10, 10, -5, 27,
grid(1),
graph(900, 900, -10, 10, -5, 27, -x^2-8x+9),
circle(-9,0,0.08),circle(-9,0,0.10),
circle(-8,9,0.08),circle(-8,9,0.10),
circle(-7,16,0.08),circle(-7,16,0.10),
circle(-6,21,0.08),circle(-6,21,0.10),
circle(-5,24,0.08),circle(-5,24,0.10),
circle(-4,25,0.08),circle(-4,25,0.10),
circle(-3,24,0.08),circle(-3,24,0.10),
circle(-2,21,0.08),circle(-2,21,0.10),
circle(-1,16,0.08),circle(-1,16,0.10),
circle(0,9,0.08),circle(0,9,0.10),
circle(1,0,0.08),circle(1,0,0.10)

)}}} Graph of {{{y=-x^2-8x+9}}}