Question 122811
# 1


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



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



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



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



{{{f(-3)=-8}}} Add -9 and 1 to get -8



So when {{{x=-3}}}, we have {{{y=-8}}}



So our 1st point is (-3,-8)



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



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



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



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



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



{{{f(-2)=-3}}} Add -4 and 1 to get -3



So when {{{x=-2}}}, we have {{{y=-3}}}



So our 2nd point is (-2,-3)



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



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



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



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



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



{{{f(-1)=0}}} Add -1 and 1 to get 0



So when {{{x=-1}}}, we have {{{y=0}}}



So our 3rd point is (-1,0)



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



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



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



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



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



{{{f(0)=1}}} Add 0 and 1 to get 1



So when {{{x=0}}}, we have {{{y=1}}}



So our 4th point is (0,1)



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



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



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



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



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



{{{f(1)=0}}} Add -1 and 1 to get 0



So when {{{x=1}}}, we have {{{y=0}}}



So our 5th point is (1,0)



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



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



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



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



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



{{{f(2)=-3}}} Add -4 and 1 to get -3



So when {{{x=2}}}, we have {{{y=-3}}}



So our 6th point is (2,-3)



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



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



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



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



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



{{{f(3)=-8}}} Add -9 and 1 to get -8



So when {{{x=3}}}, we have {{{y=-8}}}



So our 7th point is (3,-8)



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> -3</TD><TD>-8</TD></TR> 
<TR><TD> -2</TD><TD>-3</TD></TR> 
<TR><TD> -1</TD><TD>0</TD></TR> 
<TR><TD> 0</TD><TD>1</TD></TR> 
<TR><TD> 1</TD><TD>0</TD></TR> 
<TR><TD> 2</TD><TD>-3</TD></TR> 
<TR><TD> 3</TD><TD>-8</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-15,15,-15,15,
  grid( 1 ),
circle(-3,-8,0.05),
circle(-3,-8,0.08),
circle(-3,-8,0.05),
circle(-3,-8,0.1),
circle(-3,-8,0.05),
circle(-3,-8,0.12),
circle(-2,-3,0.05),
circle(-2,-3,0.08),
circle(-2,-3,0.05),
circle(-2,-3,0.1),
circle(-2,-3,0.05),
circle(-2,-3,0.12),
circle(-1,0,0.05),
circle(-1,0,0.08),
circle(-1,0,0.05),
circle(-1,0,0.1),
circle(-1,0,0.05),
circle(-1,0,0.12),
circle(0,1,0.05),
circle(0,1,0.08),
circle(0,1,0.05),
circle(0,1,0.1),
circle(0,1,0.05),
circle(0,1,0.12),
circle(1,0,0.05),
circle(1,0,0.08),
circle(1,0,0.05),
circle(1,0,0.1),
circle(1,0,0.05),
circle(1,0,0.12),
circle(2,-3,0.05),
circle(2,-3,0.08),
circle(2,-3,0.05),
circle(2,-3,0.1),
circle(2,-3,0.05),
circle(2,-3,0.12),
circle(3,-8,0.05),
circle(3,-8,0.08),
circle(3,-8,0.05),
circle(3,-8,0.1),
circle(3,-8,0.05),
circle(3,-8,0.12)
)}}}



Now connect the points to graph {{{y=-1x^2+1}}}  (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+1),
circle(-3,-8,0.05),
circle(-3,-8,0.08),
circle(-3,-8,0.05),
circle(-3,-8,0.1),
circle(-3,-8,0.05),
circle(-3,-8,0.12),
circle(-2,-3,0.05),
circle(-2,-3,0.08),
circle(-2,-3,0.05),
circle(-2,-3,0.1),
circle(-2,-3,0.05),
circle(-2,-3,0.12),
circle(-1,0,0.05),
circle(-1,0,0.08),
circle(-1,0,0.05),
circle(-1,0,0.1),
circle(-1,0,0.05),
circle(-1,0,0.12),
circle(0,1,0.05),
circle(0,1,0.08),
circle(0,1,0.05),
circle(0,1,0.1),
circle(0,1,0.05),
circle(0,1,0.12),
circle(1,0,0.05),
circle(1,0,0.08),
circle(1,0,0.05),
circle(1,0,0.1),
circle(1,0,0.05),
circle(1,0,0.12),
circle(2,-3,0.05),
circle(2,-3,0.08),
circle(2,-3,0.05),
circle(2,-3,0.1),
circle(2,-3,0.05),
circle(2,-3,0.12),
circle(3,-8,0.05),
circle(3,-8,0.08),
circle(3,-8,0.05),
circle(3,-8,0.1),
circle(3,-8,0.05),
circle(3,-8,0.12)
)}}}





<hr>



# 2



Looking at {{{y=2x}}} we can see that the equation is in slope-intercept form {{{y=mx+b}}} where the slope is {{{m=2}}} and the y-intercept is {{{b=0}}}  note: {{{y=2x}}} really looks like {{{y=2x+0}}} 



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


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


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



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

{{{slope=rise/run}}}


Also, because the slope is {{{2}}}, this means:


{{{rise/run=2/1}}}



which shows us that the rise is 2 and the run is 1. This means that to go from point to point, we can go up 2  and over 1




So starting at *[Tex \LARGE \left(0,0\right)], go up 2 units 

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


and to the right 1 unit to get to the next point *[Tex \LARGE \left(1,2\right)]

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,0,.1)),
  blue(circle(0,0,.12)),
  blue(circle(0,0,.15)),
  blue(circle(1,2,.15,1.5)),
  blue(circle(1,2,.1,1.5)),
  blue(arc(0,0+(2/2),2,2,90,270)),
  blue(arc((1/2),2,1,2, 180,360))
)}}}



Now draw a line through these points to graph {{{y=2x}}}


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  graph(500,500,-10,10,-10,10,2x),
  blue(circle(0,0,.1)),
  blue(circle(0,0,.12)),
  blue(circle(0,0,.15)),
  blue(circle(1,2,.15,1.5)),
  blue(circle(1,2,.1,1.5)),
  blue(arc(0,0+(2/2),2,2,90,270)),
  blue(arc((1/2),2,1,2, 180,360))
)}}} So this is the graph of {{{y=2x}}} through the points *[Tex \LARGE \left(0,0\right)] and *[Tex \LARGE \left(1,2\right)]