Question 83862
 In order to graph {{{-x^2+4x-1}}}, we need to plot some points. 
 So we can start at any x value. So lets start at x=-1 
 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(-1)^2+4(-1)-1}}} Plug in {{{x=-1}}} 
 
{{{-(-1)^2+4(-1)-1=-6}}} Calculate the y value

So when {{{x=-1}}} {{{y=-6}}}

So our 1st point is (-1,-6)




 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(0)^2+4(0)-1}}} Plug in {{{x=0}}} 
 
{{{-(0)^2+4(0)-1=-1}}} Calculate the y value

So when {{{x=0}}} {{{y=-1}}}

So our 2nd point is (0,-1)




 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(1)^2+4(1)-1}}} Plug in {{{x=1}}} 
 
{{{-(1)^2+4(1)-1=2}}} Calculate the y value

So when {{{x=1}}} {{{y=2}}}

So our 3rd point is (1,2)




 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(2)^2+4(2)-1}}} Plug in {{{x=2}}} 
 
{{{-(2)^2+4(2)-1=3}}} Calculate the y value

So when {{{x=2}}} {{{y=3}}}

So our 4th point is (2,3)




 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(3)^2+4(3)-1}}} Plug in {{{x=3}}} 
 
{{{-(3)^2+4(3)-1=2}}} Calculate the y value

So when {{{x=3}}} {{{y=2}}}

So our 5th point is (3,2)




 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(4)^2+4(4)-1}}} Plug in {{{x=4}}} 
 
{{{-(4)^2+4(4)-1=-1}}} Calculate the y value

So when {{{x=4}}} {{{y=-1}}}

So our 6th point is (4,-1)




 

{{{y=-x^2+4x-1}}} Start with the given equation 
 
{{{y=-(5)^2+4(5)-1}}} Plug in {{{x=5}}} 
 
{{{-(5)^2+4(5)-1=-6}}} Calculate the y value

So when {{{x=5}}} {{{y=-6}}}

So our 7th point is (5,-6)




 



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

{{{drawing( 900, 900, -10, 10, -10, 10,
  grid( 1 ),circle(-1,-6,0.05), 
  circle(-1,-6,0.08), 
circle(0,-1,0.05), 
  circle(0,-1,0.08), 
circle(1,2,0.05), 
  circle(1,2,0.08), 
circle(2,3,0.05), 
  circle(2,3,0.08), 
circle(3,2,0.05), 
  circle(3,2,0.08), 
circle(4,-1,0.05), 
  circle(4,-1,0.08), 
circle(5,-6,0.05), 
  circle(5,-6,0.08) 
)}}}

Now connect the points, this is the graph of {{{-x^2+4x-1}}}

{{{drawing( 900, 900, -10, 10, -10, 10,
  grid( 1 ),
  graph( 900, 900,-10, 10, -10, 10, -x^2+4x-1),circle(-1,-6,0.05), 
  circle(-1,-6,0.08), 
circle(0,-1,0.05), 
  circle(0,-1,0.08), 
circle(1,2,0.05), 
  circle(1,2,0.08), 
circle(2,3,0.05), 
  circle(2,3,0.08), 
circle(3,2,0.05), 
  circle(3,2,0.08), 
circle(4,-1,0.05), 
  circle(4,-1,0.08), 
circle(5,-6,0.05), 
  circle(5,-6,0.08) 
)}}}


So from the graph we can see that the vertex is (2,3), the domain is the element of all real numbers (any number will work for x) and the range is {{{y<=3}}}