Question 128757
In order to graph {{{f(x)=-1x^2+6x+5}}}, we need to plot some points. To do that, we need to plug in some x values to get some y values



So let's find the first point:




{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(-2)=-1(-2)^2+6(-2)+5}}} Plug in {{{x=-2}}}



{{{f(-2)=-1*4+6*-2+5}}} Raise -2 to the 2nd power to get 4



{{{f(-2)=-4+6*-2+5}}} Multiply -1 and 4 to get -4



{{{f(-2)=-4+-12+5}}} Multiply 6 and -2 to get -12



{{{f(-2)=-16+5}}} Add -4 and -12 to get -16



{{{f(-2)=-11}}} Add -16 and 5 to get -11



So when {{{x=-2}}}, we have {{{y=-11}}}



So our 1st point is (-2,-11)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(-1)=-1(-1)^2+6(-1)+5}}} Plug in {{{x=-1}}}



{{{f(-1)=-1*1+6*-1+5}}} Raise -1 to the 2nd power to get 1



{{{f(-1)=-1+6*-1+5}}} Multiply -1 and 1 to get -1



{{{f(-1)=-1+-6+5}}} Multiply 6 and -1 to get -6



{{{f(-1)=-7+5}}} Add -1 and -6 to get -7



{{{f(-1)=-2}}} Add -7 and 5 to get -2



So when {{{x=-1}}}, we have {{{y=-2}}}



So our 2nd point is (-1,-2)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(0)=-1(0)^2+6(0)+5}}} Plug in {{{x=0}}}



{{{f(0)=-1*0+6*0+5}}} Raise 0 to the 2nd power to get 0



{{{f(0)=0+6*0+5}}} Multiply -1 and 0 to get 0



{{{f(0)=0+0+5}}} Multiply 6 and 0 to get 0



{{{f(0)=0+5}}} Add 0 and 0 to get 0



{{{f(0)=5}}} Add 0 and 5 to get 5



So when {{{x=0}}}, we have {{{y=5}}}



So our 3rd point is (0,5)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(1)=-1(1)^2+6(1)+5}}} Plug in {{{x=1}}}



{{{f(1)=-1*1+6*1+5}}} Raise 1 to the 2nd power to get 1



{{{f(1)=-1+6*1+5}}} Multiply -1 and 1 to get -1



{{{f(1)=-1+6+5}}} Multiply 6 and 1 to get 6



{{{f(1)=5+5}}} Add -1 and 6 to get 5



{{{f(1)=10}}} Add 5 and 5 to get 10



So when {{{x=1}}}, we have {{{y=10}}}



So our 4th point is (1,10)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(2)=-1(2)^2+6(2)+5}}} Plug in {{{x=2}}}



{{{f(2)=-1*4+6*2+5}}} Raise 2 to the 2nd power to get 4



{{{f(2)=-4+6*2+5}}} Multiply -1 and 4 to get -4



{{{f(2)=-4+12+5}}} Multiply 6 and 2 to get 12



{{{f(2)=8+5}}} Add -4 and 12 to get 8



{{{f(2)=13}}} Add 8 and 5 to get 13



So when {{{x=2}}}, we have {{{y=13}}}



So our 5th point is (2,13)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(3)=-1(3)^2+6(3)+5}}} Plug in {{{x=3}}}



{{{f(3)=-1*9+6*3+5}}} Raise 3 to the 2nd power to get 9



{{{f(3)=-9+6*3+5}}} Multiply -1 and 9 to get -9



{{{f(3)=-9+18+5}}} Multiply 6 and 3 to get 18



{{{f(3)=9+5}}} Add -9 and 18 to get 9



{{{f(3)=14}}} Add 9 and 5 to get 14



So when {{{x=3}}}, we have {{{y=14}}}



So our 6th point is (3,14)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(4)=-1(4)^2+6(4)+5}}} Plug in {{{x=4}}}



{{{f(4)=-1*16+6*4+5}}} Raise 4 to the 2nd power to get 16



{{{f(4)=-16+6*4+5}}} Multiply -1 and 16 to get -16



{{{f(4)=-16+24+5}}} Multiply 6 and 4 to get 24



{{{f(4)=8+5}}} Add -16 and 24 to get 8



{{{f(4)=13}}} Add 8 and 5 to get 13



So when {{{x=4}}}, we have {{{y=13}}}



So our 7th point is (4,13)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(5)=-1(5)^2+6(5)+5}}} Plug in {{{x=5}}}



{{{f(5)=-1*25+6*5+5}}} Raise 5 to the 2nd power to get 25



{{{f(5)=-25+6*5+5}}} Multiply -1 and 25 to get -25



{{{f(5)=-25+30+5}}} Multiply 6 and 5 to get 30



{{{f(5)=5+5}}} Add -25 and 30 to get 5



{{{f(5)=10}}} Add 5 and 5 to get 10



So when {{{x=5}}}, we have {{{y=10}}}



So our 8th point is (5,10)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(6)=-1(6)^2+6(6)+5}}} Plug in {{{x=6}}}



{{{f(6)=-1*36+6*6+5}}} Raise 6 to the 2nd power to get 36



{{{f(6)=-36+6*6+5}}} Multiply -1 and 36 to get -36



{{{f(6)=-36+36+5}}} Multiply 6 and 6 to get 36



{{{f(6)=0+5}}} Add -36 and 36 to get 0



{{{f(6)=5}}} Add 0 and 5 to get 5



So when {{{x=6}}}, we have {{{y=5}}}



So our 9th point is (6,5)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(7)=-1(7)^2+6(7)+5}}} Plug in {{{x=7}}}



{{{f(7)=-1*49+6*7+5}}} Raise 7 to the 2nd power to get 49



{{{f(7)=-49+6*7+5}}} Multiply -1 and 49 to get -49



{{{f(7)=-49+42+5}}} Multiply 6 and 7 to get 42



{{{f(7)=-7+5}}} Add -49 and 42 to get -7



{{{f(7)=-2}}} Add -7 and 5 to get -2



So when {{{x=7}}}, we have {{{y=-2}}}



So our 10th point is (7,-2)



--------------  Let's find another point  --------------



{{{f(x)=-1x^2+6x+5}}} Start with the given function



{{{f(8)=-1(8)^2+6(8)+5}}} Plug in {{{x=8}}}



{{{f(8)=-1*64+6*8+5}}} Raise 8 to the 2nd power to get 64



{{{f(8)=-64+6*8+5}}} Multiply -1 and 64 to get -64



{{{f(8)=-64+48+5}}} Multiply 6 and 8 to get 48



{{{f(8)=-16+5}}} Add -64 and 48 to get -16



{{{f(8)=-11}}} Add -16 and 5 to get -11



So when {{{x=8}}}, we have {{{y=-11}}}



So our 11th point is (8,-11)



Now lets make a table of the values we have calculated

<pre>
<TABLE width=500>

<TR><TD> x</TD><TD>y</TD></TR>

<TR><TD> -2</TD><TD>-11</TD></TR> 
<TR><TD> -1</TD><TD>-2</TD></TR> 
<TR><TD> 0</TD><TD>5</TD></TR> 
<TR><TD> 1</TD><TD>10</TD></TR> 
<TR><TD> 2</TD><TD>13</TD></TR> 
<TR><TD> 3</TD><TD>14</TD></TR> 
<TR><TD> 4</TD><TD>13</TD></TR> 
<TR><TD> 5</TD><TD>10</TD></TR> 
<TR><TD> 6</TD><TD>5</TD></TR> 
<TR><TD> 7</TD><TD>-2</TD></TR> 
<TR><TD> 8</TD><TD>-11</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-15,15,-15,15,
  grid( 1 ),
circle(-2,-11,0.05),
circle(-2,-11,0.08),
circle(-2,-11,0.05),
circle(-2,-11,0.1),
circle(-2,-11,0.05),
circle(-2,-11,0.12),
circle(-1,-2,0.05),
circle(-1,-2,0.08),
circle(-1,-2,0.05),
circle(-1,-2,0.1),
circle(-1,-2,0.05),
circle(-1,-2,0.12),
circle(0,5,0.05),
circle(0,5,0.08),
circle(0,5,0.05),
circle(0,5,0.1),
circle(0,5,0.05),
circle(0,5,0.12),
circle(1,10,0.05),
circle(1,10,0.08),
circle(1,10,0.05),
circle(1,10,0.1),
circle(1,10,0.05),
circle(1,10,0.12),
circle(2,13,0.05),
circle(2,13,0.08),
circle(2,13,0.05),
circle(2,13,0.1),
circle(2,13,0.05),
circle(2,13,0.12),
circle(3,14,0.05),
circle(3,14,0.08),
circle(3,14,0.05),
circle(3,14,0.1),
circle(3,14,0.05),
circle(3,14,0.12),
circle(4,13,0.05),
circle(4,13,0.08),
circle(4,13,0.05),
circle(4,13,0.1),
circle(4,13,0.05),
circle(4,13,0.12),
circle(5,10,0.05),
circle(5,10,0.08),
circle(5,10,0.05),
circle(5,10,0.1),
circle(5,10,0.05),
circle(5,10,0.12),
circle(6,5,0.05),
circle(6,5,0.08),
circle(6,5,0.05),
circle(6,5,0.1),
circle(6,5,0.05),
circle(6,5,0.12),
circle(7,-2,0.05),
circle(7,-2,0.08),
circle(7,-2,0.05),
circle(7,-2,0.1),
circle(7,-2,0.05),
circle(7,-2,0.12),
circle(8,-11,0.05),
circle(8,-11,0.08),
circle(8,-11,0.05),
circle(8,-11,0.1),
circle(8,-11,0.05),
circle(8,-11,0.12)
)}}}



Now connect the points to graph {{{y=-1x^2+6x+5}}}  (note: the more points you plot, the easier it is to draw the graph)

{{{drawing(900,900,-15,15,-15,15,
grid( 1 ),
graph(900,900,-15,15,-15,15, -1x^2+6x+5),
circle(-2,-11,0.05),
circle(-2,-11,0.08),
circle(-2,-11,0.05),
circle(-2,-11,0.1),
circle(-2,-11,0.05),
circle(-2,-11,0.12),
circle(-1,-2,0.05),
circle(-1,-2,0.08),
circle(-1,-2,0.05),
circle(-1,-2,0.1),
circle(-1,-2,0.05),
circle(-1,-2,0.12),
circle(0,5,0.05),
circle(0,5,0.08),
circle(0,5,0.05),
circle(0,5,0.1),
circle(0,5,0.05),
circle(0,5,0.12),
circle(1,10,0.05),
circle(1,10,0.08),
circle(1,10,0.05),
circle(1,10,0.1),
circle(1,10,0.05),
circle(1,10,0.12),
circle(2,13,0.05),
circle(2,13,0.08),
circle(2,13,0.05),
circle(2,13,0.1),
circle(2,13,0.05),
circle(2,13,0.12),
circle(3,14,0.05),
circle(3,14,0.08),
circle(3,14,0.05),
circle(3,14,0.1),
circle(3,14,0.05),
circle(3,14,0.12),
circle(4,13,0.05),
circle(4,13,0.08),
circle(4,13,0.05),
circle(4,13,0.1),
circle(4,13,0.05),
circle(4,13,0.12),
circle(5,10,0.05),
circle(5,10,0.08),
circle(5,10,0.05),
circle(5,10,0.1),
circle(5,10,0.05),
circle(5,10,0.12),
circle(6,5,0.05),
circle(6,5,0.08),
circle(6,5,0.05),
circle(6,5,0.1),
circle(6,5,0.05),
circle(6,5,0.12),
circle(7,-2,0.05),
circle(7,-2,0.08),
circle(7,-2,0.05),
circle(7,-2,0.1),
circle(7,-2,0.05),
circle(7,-2,0.12),
circle(8,-11,0.05),
circle(8,-11,0.08),
circle(8,-11,0.05),
circle(8,-11,0.1),
circle(8,-11,0.05),
circle(8,-11,0.12)
)}}}