Question 122812
# 1


In order to graph {{{f(x)=1x^2-4x}}}, 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^2-4x}}} Start with the given function



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



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



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



{{{f(-2)=4--8}}} Multiply 4 and -2 to get -8



{{{f(-2)=12}}} Subtract -8 from 4 to get 12



So when {{{x=-2}}}, we have {{{y=12}}}



So our 1st point is (-2,12)



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



{{{f(x)=1x^2-4x}}} Start with the given function



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



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



{{{f(-1)=1-4*-1}}} Multiply 1 and 1 to get 1



{{{f(-1)=1--4}}} Multiply 4 and -1 to get -4



{{{f(-1)=5}}} Subtract -4 from 1 to get 5



So when {{{x=-1}}}, we have {{{y=5}}}



So our 2nd point is (-1,5)



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



{{{f(x)=1x^2-4x}}} Start with the given function



{{{f(0)=1(0)^2-4(0)}}} Plug in {{{x=0}}}



{{{f(0)=1*0-4*0}}} Raise 0 to the 2nd power to get 0



{{{f(0)=0-4*0}}} Multiply 1 and 0 to get 0



{{{f(0)=0-0}}} Multiply 4 and 0 to get 0



{{{f(0)=0}}} Subtract 0 from 0 to get 0



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



So our 3rd point is (0,0)



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



{{{f(x)=1x^2-4x}}} Start with the given function



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



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



{{{f(1)=1-4*1}}} Multiply 1 and 1 to get 1



{{{f(1)=1-4}}} Multiply 4 and 1 to get 4



{{{f(1)=-3}}} Subtract 4 from 1 to get -3



So when {{{x=1}}}, we have {{{y=-3}}}



So our 4th point is (1,-3)



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



{{{f(x)=1x^2-4x}}} Start with the given function



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



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



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



{{{f(2)=4-8}}} Multiply 4 and 2 to get 8



{{{f(2)=-4}}} Subtract 8 from 4 to get -4



So when {{{x=2}}}, we have {{{y=-4}}}



So our 5th point is (2,-4)



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



{{{f(x)=1x^2-4x}}} Start with the given function



{{{f(3)=1(3)^2-4(3)}}} Plug in {{{x=3}}}



{{{f(3)=1*9-4*3}}} Raise 3 to the 2nd power to get 9



{{{f(3)=9-4*3}}} Multiply 1 and 9 to get 9



{{{f(3)=9-12}}} Multiply 4 and 3 to get 12



{{{f(3)=-3}}} Subtract 12 from 9 to get -3



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



So our 6th point is (3,-3)



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



{{{f(x)=1x^2-4x}}} Start with the given function



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



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



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



{{{f(4)=16-16}}} Multiply 4 and 4 to get 16



{{{f(4)=0}}} Subtract 16 from 16 to get 0



So when {{{x=4}}}, we have {{{y=0}}}



So our 7th point is (4,0)



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



{{{f(x)=1x^2-4x}}} Start with the given function



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



{{{f(5)=1*25-4*5}}} Raise 5 to the 2nd power to get 25



{{{f(5)=25-4*5}}} Multiply 1 and 25 to get 25



{{{f(5)=25-20}}} Multiply 4 and 5 to get 20



{{{f(5)=5}}} Subtract 20 from 25 to get 5



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



So our 8th point is (5,5)



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



{{{f(x)=1x^2-4x}}} Start with the given function



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



{{{f(6)=1*36-4*6}}} Raise 6 to the 2nd power to get 36



{{{f(6)=36-4*6}}} Multiply 1 and 36 to get 36



{{{f(6)=36-24}}} Multiply 4 and 6 to get 24



{{{f(6)=12}}} Subtract 24 from 36 to get 12



So when {{{x=6}}}, we have {{{y=12}}}



So our 9th point is (6,12)



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>12</TD></TR> 
<TR><TD> -1</TD><TD>5</TD></TR> 
<TR><TD> 0</TD><TD>0</TD></TR> 
<TR><TD> 1</TD><TD>-3</TD></TR> 
<TR><TD> 2</TD><TD>-4</TD></TR> 
<TR><TD> 3</TD><TD>-3</TD></TR> 
<TR><TD> 4</TD><TD>0</TD></TR> 
<TR><TD> 5</TD><TD>5</TD></TR> 
<TR><TD> 6</TD><TD>12</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-15,15,-15,15,
  grid( 1 ),
circle(-2,12,0.05),
circle(-2,12,0.08),
circle(-2,12,0.05),
circle(-2,12,0.1),
circle(-2,12,0.05),
circle(-2,12,0.12),
circle(-1,5,0.05),
circle(-1,5,0.08),
circle(-1,5,0.05),
circle(-1,5,0.1),
circle(-1,5,0.05),
circle(-1,5,0.12),
circle(0,0,0.05),
circle(0,0,0.08),
circle(0,0,0.05),
circle(0,0,0.1),
circle(0,0,0.05),
circle(0,0,0.12),
circle(1,-3,0.05),
circle(1,-3,0.08),
circle(1,-3,0.05),
circle(1,-3,0.1),
circle(1,-3,0.05),
circle(1,-3,0.12),
circle(2,-4,0.05),
circle(2,-4,0.08),
circle(2,-4,0.05),
circle(2,-4,0.1),
circle(2,-4,0.05),
circle(2,-4,0.12),
circle(3,-3,0.05),
circle(3,-3,0.08),
circle(3,-3,0.05),
circle(3,-3,0.1),
circle(3,-3,0.05),
circle(3,-3,0.12),
circle(4,0,0.05),
circle(4,0,0.08),
circle(4,0,0.05),
circle(4,0,0.1),
circle(4,0,0.05),
circle(4,0,0.12),
circle(5,5,0.05),
circle(5,5,0.08),
circle(5,5,0.05),
circle(5,5,0.1),
circle(5,5,0.05),
circle(5,5,0.12),
circle(6,12,0.05),
circle(6,12,0.08),
circle(6,12,0.05),
circle(6,12,0.1),
circle(6,12,0.05),
circle(6,12,0.12)
)}}}



Now connect the points to graph {{{y=1x^2-4x}}}  (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^2-4x),
circle(-2,12,0.05),
circle(-2,12,0.08),
circle(-2,12,0.05),
circle(-2,12,0.1),
circle(-2,12,0.05),
circle(-2,12,0.12),
circle(-1,5,0.05),
circle(-1,5,0.08),
circle(-1,5,0.05),
circle(-1,5,0.1),
circle(-1,5,0.05),
circle(-1,5,0.12),
circle(0,0,0.05),
circle(0,0,0.08),
circle(0,0,0.05),
circle(0,0,0.1),
circle(0,0,0.05),
circle(0,0,0.12),
circle(1,-3,0.05),
circle(1,-3,0.08),
circle(1,-3,0.05),
circle(1,-3,0.1),
circle(1,-3,0.05),
circle(1,-3,0.12),
circle(2,-4,0.05),
circle(2,-4,0.08),
circle(2,-4,0.05),
circle(2,-4,0.1),
circle(2,-4,0.05),
circle(2,-4,0.12),
circle(3,-3,0.05),
circle(3,-3,0.08),
circle(3,-3,0.05),
circle(3,-3,0.1),
circle(3,-3,0.05),
circle(3,-3,0.12),
circle(4,0,0.05),
circle(4,0,0.08),
circle(4,0,0.05),
circle(4,0,0.1),
circle(4,0,0.05),
circle(4,0,0.12),
circle(5,5,0.05),
circle(5,5,0.08),
circle(5,5,0.05),
circle(5,5,0.1),
circle(5,5,0.05),
circle(5,5,0.12),
circle(6,12,0.05),
circle(6,12,0.08),
circle(6,12,0.05),
circle(6,12,0.1),
circle(6,12,0.05),
circle(6,12,0.12)
)}}}





<hr>




# 2






Looking at {{{y=-x+1}}} we can see that the equation is in slope-intercept form {{{y=mx+b}}} where the slope is {{{m=-1}}} and the y-intercept is {{{b=1}}} 



Since {{{b=1}}} this tells us that the y-intercept is *[Tex \LARGE \left(0,1\right)].Remember the y-intercept is the point where the graph intersects with the y-axis


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


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,1,.1)),
  blue(circle(0,1,.12)),
  blue(circle(0,1,.15))
)}}}



Now since the slope is comprised of the "rise" over the "run" this means

{{{slope=rise/run}}}


Also, because the slope is {{{-1}}}, this means:


{{{rise/run=-1/1}}}



which shows us that the rise is -1 and the run is 1. This means that to go from point to point, we can go down 1  and over 1




So starting at *[Tex \LARGE \left(0,1\right)], go down 1 unit 

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,1,.1)),
  blue(circle(0,1,.12)),
  blue(circle(0,1,.15)),
  blue(arc(0,1+(-1/2),2,-1,90,270))
)}}}


and to the right 1 unit to get to the next point *[Tex \LARGE \left(1,0\right)]

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,1,.1)),
  blue(circle(0,1,.12)),
  blue(circle(0,1,.15)),
  blue(circle(1,0,.15,1.5)),
  blue(circle(1,0,.1,1.5)),
  blue(arc(0,1+(-1/2),2,-1,90,270)),
  blue(arc((1/2),0,1,2, 0,180))
)}}}



Now draw a line through these points to graph {{{y=-x+1}}}


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  graph(500,500,-10,10,-10,10,-x+1),
  blue(circle(0,1,.1)),
  blue(circle(0,1,.12)),
  blue(circle(0,1,.15)),
  blue(circle(1,0,.15,1.5)),
  blue(circle(1,0,.1,1.5)),
  blue(arc(0,1+(-1/2),2,-1,90,270)),
  blue(arc((1/2),0,1,2, 0,180))
)}}} So this is the graph of {{{y=-x+1}}} through the points *[Tex \LARGE \left(0,1\right)] and *[Tex \LARGE \left(1,0\right)]