<PRE><B>Question 188748</B>
# 2


c)


In order to graph {{{y=x^2-6x-7}}}, we need to plot a few points.



Let&#39;s find the y value when {{{x=-1}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(-1)^2-6(-1)-7}}} Plug in {{{x=-1}}}.



{{{y=1-6(-1)-7}}} Square -1 to get 1.



{{{y=1+6-7}}} Multiply -6 and -1 to get 6.



{{{y=0}}} Combine like terms.



So when {{{x=-1}}}, {{{y=0}}}. So we have the point (-1,0).



----------------------------



Let&#39;s find the y value when {{{x=0}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(0)^2-6(0)-7}}} Plug in {{{x=0}}}.



{{{y=0-6(0)-7}}} Square 0 to get 0.



{{{y=0+0-7}}} Multiply -6 and 0 to get 0.



{{{y=-7}}} Combine like terms.



So when {{{x=0}}}, {{{y=-7}}}. So we have the point (0,-7).



----------------------------



Let&#39;s find the y value when {{{x=1}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(1)^2-6(1)-7}}} Plug in {{{x=1}}}.



{{{y=1-6(1)-7}}} Square 1 to get 1.



{{{y=1-6-7}}} Multiply -6 and 1 to get -6.



{{{y=-12}}} Combine like terms.



So when {{{x=1}}}, {{{y=-12}}}. So we have the point (1,-12).



----------------------------



Let&#39;s find the y value when {{{x=2}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(2)^2-6(2)-7}}} Plug in {{{x=2}}}.



{{{y=4-6(2)-7}}} Square 2 to get 4.



{{{y=4-12-7}}} Multiply -6 and 2 to get -12.



{{{y=-15}}} Combine like terms.



So when {{{x=2}}}, {{{y=-15}}}. So we have the point (2,-15).



----------------------------



Let&#39;s find the y value when {{{x=3}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(3)^2-6(3)-7}}} Plug in {{{x=3}}}.



{{{y=9-6(3)-7}}} Square 3 to get 9.



{{{y=9-18-7}}} Multiply -6 and 3 to get -18.



{{{y=-16}}} Combine like terms.



So when {{{x=3}}}, {{{y=-16}}}. So we have the point (3,-16).



----------------------------



Let&#39;s find the y value when {{{x=4}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(4)^2-6(4)-7}}} Plug in {{{x=4}}}.



{{{y=16-6(4)-7}}} Square 4 to get 16.



{{{y=16-24-7}}} Multiply -6 and 4 to get -24.



{{{y=-15}}} Combine like terms.



So when {{{x=4}}}, {{{y=-15}}}. So we have the point (4,-15).



----------------------------



Let&#39;s find the y value when {{{x=5}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(5)^2-6(5)-7}}} Plug in {{{x=5}}}.



{{{y=25-6(5)-7}}} Square 5 to get 25.



{{{y=25-30-7}}} Multiply -6 and 5 to get -30.



{{{y=-12}}} Combine like terms.



So when {{{x=5}}}, {{{y=-12}}}. So we have the point (5,-12).



----------------------------



Let&#39;s find the y value when {{{x=6}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(6)^2-6(6)-7}}} Plug in {{{x=6}}}.



{{{y=36-6(6)-7}}} Square 6 to get 36.



{{{y=36-36-7}}} Multiply -6 and 6 to get -36.



{{{y=-7}}} Combine like terms.



So when {{{x=6}}}, {{{y=-7}}}. So we have the point (6,-7).



----------------------------



Let&#39;s find the y value when {{{x=7}}}



{{{y=x^2-6x-7}}} Start with the given equation.



{{{y=(7)^2-6(7)-7}}} Plug in {{{x=7}}}.



{{{y=49-6(7)-7}}} Square 7 to get 49.



{{{y=49-42-7}}} Multiply -6 and 7 to get -42.



{{{y=0}}} Combine like terms.



So when {{{x=7}}}, {{{y=0}}}. So we have the point (7,0).



----------------------------



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;0&lt;/td&gt;&lt;td&gt;-7&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;-12&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;-15&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;3&lt;/td&gt;&lt;td&gt;-16&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;-15&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;5&lt;/td&gt;&lt;td&gt;-12&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;6&lt;/td&gt;&lt;td&gt;-7&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;7&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;

Now let&#39;s plot the points:



{{{ drawing(500, 500, -5, 13, -20, 3,
grid(1),
graph(500, 500, -5, 13, -20, 3, 0),
circle(-1,0,0.08),circle(-1,0,0.10),
circle(0,-7,0.08),circle(0,-7,0.10),
circle(1,-12,0.08),circle(1,-12,0.10),
circle(2,-15,0.08),circle(2,-15,0.10),
circle(3,-16,0.08),circle(3,-16,0.10),
circle(4,-15,0.08),circle(4,-15,0.10),
circle(5,-12,0.08),circle(5,-12,0.10),
circle(6,-7,0.08),circle(6,-7,0.10),
circle(7,0,0.08),circle(7,0,0.10)

)}}}


Now draw a curve through all of the points to graph {{{y=x^2-6x-7}}}:



{{{ drawing(500, 500, -5, 13, -20, 3,
grid(1),
graph(500, 500, -5, 13, -20, 3, x^2-6x-7),
circle(-1,0,0.08),circle(-1,0,0.10),
circle(0,-7,0.08),circle(0,-7,0.10),
circle(1,-12,0.08),circle(1,-12,0.10),
circle(2,-15,0.08),circle(2,-15,0.10),
circle(3,-16,0.08),circle(3,-16,0.10),
circle(4,-15,0.08),circle(4,-15,0.10),
circle(5,-12,0.08),circle(5,-12,0.10),
circle(6,-7,0.08),circle(6,-7,0.10),
circle(7,0,0.08),circle(7,0,0.10)

)}}} Graph of {{{y=x^2-6x-7}}}



-------------------------------------------


# 2


d)


In order to graph {{{y=x^2-4x-5}}}, we need to plot a few points.



Let&#39;s find the y value when {{{x=-2}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(-2)^2-4(-2)-5}}} Plug in {{{x=-2}}}.



{{{y=4-4(-2)-5}}} Square -2 to get 4.



{{{y=4+8-5}}} Multiply -4 and -2 to get 8.



{{{y=7}}} Combine like terms.



So when {{{x=-2}}}, {{{y=7}}}. So we have the point (-2,7).



----------------------------



Let&#39;s find the y value when {{{x=-1}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(-1)^2-4(-1)-5}}} Plug in {{{x=-1}}}.



{{{y=1-4(-1)-5}}} Square -1 to get 1.



{{{y=1+4-5}}} Multiply -4 and -1 to get 4.



{{{y=0}}} Combine like terms.



So when {{{x=-1}}}, {{{y=0}}}. So we have the point (-1,0).



----------------------------



Let&#39;s find the y value when {{{x=0}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(0)^2-4(0)-5}}} Plug in {{{x=0}}}.



{{{y=0-4(0)-5}}} Square 0 to get 0.



{{{y=0+0-5}}} Multiply -4 and 0 to get 0.



{{{y=-5}}} Combine like terms.



So when {{{x=0}}}, {{{y=-5}}}. So we have the point (0,-5).



----------------------------



Let&#39;s find the y value when {{{x=1}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(1)^2-4(1)-5}}} Plug in {{{x=1}}}.



{{{y=1-4(1)-5}}} Square 1 to get 1.



{{{y=1-4-5}}} Multiply -4 and 1 to get -4.



{{{y=-8}}} Combine like terms.



So when {{{x=1}}}, {{{y=-8}}}. So we have the point (1,-8).



----------------------------



Let&#39;s find the y value when {{{x=2}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(2)^2-4(2)-5}}} Plug in {{{x=2}}}.



{{{y=4-4(2)-5}}} Square 2 to get 4.



{{{y=4-8-5}}} Multiply -4 and 2 to get -8.



{{{y=-9}}} Combine like terms.



So when {{{x=2}}}, {{{y=-9}}}. So we have the point (2,-9).



----------------------------



Let&#39;s find the y value when {{{x=3}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(3)^2-4(3)-5}}} Plug in {{{x=3}}}.



{{{y=9-4(3)-5}}} Square 3 to get 9.



{{{y=9-12-5}}} Multiply -4 and 3 to get -12.



{{{y=-8}}} Combine like terms.



So when {{{x=3}}}, {{{y=-8}}}. So we have the point (3,-8).



----------------------------



Let&#39;s find the y value when {{{x=4}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(4)^2-4(4)-5}}} Plug in {{{x=4}}}.



{{{y=16-4(4)-5}}} Square 4 to get 16.



{{{y=16-16-5}}} Multiply -4 and 4 to get -16.



{{{y=-5}}} Combine like terms.



So when {{{x=4}}}, {{{y=-5}}}. So we have the point (4,-5).



----------------------------



Let&#39;s find the y value when {{{x=5}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(5)^2-4(5)-5}}} Plug in {{{x=5}}}.



{{{y=25-4(5)-5}}} Square 5 to get 25.



{{{y=25-20-5}}} Multiply -4 and 5 to get -20.



{{{y=0}}} Combine like terms.



So when {{{x=5}}}, {{{y=0}}}. So we have the point (5,0).



----------------------------



Let&#39;s find the y value when {{{x=6}}}



{{{y=x^2-4x-5}}} Start with the given equation.



{{{y=(6)^2-4(6)-5}}} Plug in {{{x=6}}}.



{{{y=36-4(6)-5}}} Square 6 to get 36.



{{{y=36-24-5}}} Multiply -4 and 6 to get -24.



{{{y=7}}} Combine like terms.



So when {{{x=6}}}, {{{y=7}}}. So we have the point (6,7).



----------------------------



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;-2&lt;/td&gt;&lt;td&gt;7&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;0&lt;/td&gt;&lt;td&gt;-5&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;-8&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;2&lt;/td&gt;&lt;td&gt;-9&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;3&lt;/td&gt;&lt;td&gt;-8&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;4&lt;/td&gt;&lt;td&gt;-5&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;5&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;6&lt;/td&gt;&lt;td&gt;7&lt;/td&gt;&lt;/tr&gt;
&lt;/TABLE&gt;

&lt;/pre&gt;

Now let&#39;s plot the points:



{{{ drawing(500, 500, -10, 10, -10, 10,
grid(1),
graph(500, 500, -10, 10, -10, 10, 0),
circle(-2,7,0.08),circle(-2,7,0.10),
circle(-1,0,0.08),circle(-1,0,0.10),
circle(0,-5,0.08),circle(0,-5,0.10),
circle(1,-8,0.08),circle(1,-8,0.10),
circle(2,-9,0.08),circle(2,-9,0.10),
circle(3,-8,0.08),circle(3,-8,0.10),
circle(4,-5,0.08),circle(4,-5,0.10),
circle(5,0,0.08),circle(5,0,0.10),
circle(6,7,0.08),circle(6,7,0.10)

)}}}


Now draw a curve through all of the points to graph {{{y=x^2-4x-5}}}:



{{{ drawing(500, 500, -10, 10, -10, 10,
grid(1),
graph(500, 500, -10, 10, -10, 10, x^2-4x-5),
circle(-2,7,0.08),circle(-2,7,0.10),
circle(-1,0,0.08),circle(-1,0,0.10),
circle(0,-5,0.08),circle(0,-5,0.10),
circle(1,-8,0.08),circle(1,-8,0.10),
circle(2,-9,0.08),circle(2,-9,0.10),
circle(3,-8,0.08),circle(3,-8,0.10),
circle(4,-5,0.08),circle(4,-5,0.10),
circle(5,0,0.08),circle(5,0,0.10),
circle(6,7,0.08),circle(6,7,0.10)

)}}} Graph of {{{y=x^2-4x-5}}}</PRE>