Question 211263
First, let's graph {{{y=x^2-4x}}}



In order to graph {{{y=x^2-4x}}}, we need to plot a few points.



To get points in the form of (x,y), we need to find corresponding 'y' values to given 'x' values.



Let's find the y value when {{{x=-2}}} note: you can start at any x value. 



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(-2)^2-4(-2)}}} Plug in {{{x=-2}}}.



{{{y=4-4(-2)}}} Square {{{-2}}} to get {{{4}}}



{{{y=4--8}}} Multiply



{{{y=12}}} Combine like terms.



So if {{{x=-2}}}, then {{{y=12}}} which gives us the point (-2,12).



----------------------------



Let's find the y value when {{{x=-1}}}  



{{{y=x^2-4x}}} Start with the given equation.



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



{{{y=1-4(-1)}}} Square {{{-1}}} to get {{{1}}}



{{{y=1--4}}} Multiply



{{{y=5}}} Combine like terms.



So if {{{x=-1}}}, then {{{y=5}}} which gives us the point (-1,5).



----------------------------



Let's find the y value when {{{x=0}}}  



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(0)^2-4(0)}}} Plug in {{{x=0}}}.



{{{y=0-4(0)}}} Square {{{0}}} to get {{{0}}}



{{{y=0-0}}} Multiply



{{{y=0}}} Combine like terms.



So if {{{x=0}}}, then {{{y=0}}} which gives us the point (0,0).



----------------------------



Let's find the y value when {{{x=1}}}  



{{{y=x^2-4x}}} Start with the given equation.



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



{{{y=1-4(1)}}} Square {{{1}}} to get {{{1}}}



{{{y=1-4}}} Multiply



{{{y=-3}}} Combine like terms.



So if {{{x=1}}}, then {{{y=-3}}} which gives us the point (1,-3).



----------------------------



Let's find the y value when {{{x=2}}}  



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(2)^2-4(2)}}} Plug in {{{x=2}}}.



{{{y=4-4(2)}}} Square {{{2}}} to get {{{4}}}



{{{y=4-8}}} Multiply



{{{y=-4}}} Combine like terms.



So if {{{x=2}}}, then {{{y=-4}}} which gives us the point (2,-4).



----------------------------



Let's find the y value when {{{x=3}}}  



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(3)^2-4(3)}}} Plug in {{{x=3}}}.



{{{y=9-4(3)}}} Square {{{3}}} to get {{{9}}}



{{{y=9-12}}} Multiply



{{{y=-3}}} Combine like terms.



So if {{{x=3}}}, then {{{y=-3}}} which gives us the point (3,-3).



----------------------------



Let's find the y value when {{{x=4}}}  



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(4)^2-4(4)}}} Plug in {{{x=4}}}.



{{{y=16-4(4)}}} Square {{{4}}} to get {{{16}}}



{{{y=16-16}}} Multiply



{{{y=0}}} Combine like terms.



So if {{{x=4}}}, then {{{y=0}}} which gives us the point (4,0).



----------------------------



Let's find the y value when {{{x=5}}}  



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(5)^2-4(5)}}} Plug in {{{x=5}}}.



{{{y=25-4(5)}}} Square {{{5}}} to get {{{25}}}



{{{y=25-20}}} Multiply



{{{y=5}}} Combine like terms.



So if {{{x=5}}}, then {{{y=5}}} which gives us the point (5,5).



----------------------------



Let's find the y value when {{{x=6}}}  



{{{y=x^2-4x}}} Start with the given equation.



{{{y=(6)^2-4(6)}}} Plug in {{{x=6}}}.



{{{y=36-4(6)}}} Square {{{6}}} to get {{{36}}}



{{{y=36-24}}} Multiply



{{{y=12}}} Combine like terms.



So if {{{x=6}}}, then {{{y=12}}} which gives us the point (6,12).



----------------------------



Now let's make a table of the values we just found.



<h4>Table of Values:</h4><pre>

<TABLE border="1" width="100">
<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 let's plot the points:



{{{ drawing(500, 500, -10, 10, -5, 15,
grid(1),
graph(500, 500, -10, 10, -5, 15, 0),
circle(-2,12,0.08),circle(-2,12,0.10),circle(-2,12,0.12),circle(-2,12,0.14),
circle(-1,5,0.08),circle(-1,5,0.10),circle(-1,5,0.12),circle(-1,5,0.14),
circle(0,0,0.08),circle(0,0,0.10),circle(0,0,0.12),circle(0,0,0.14),
circle(1,-3,0.08),circle(1,-3,0.10),circle(1,-3,0.12),circle(1,-3,0.14),
circle(2,-4,0.08),circle(2,-4,0.10),circle(2,-4,0.12),circle(2,-4,0.14),
circle(3,-3,0.08),circle(3,-3,0.10),circle(3,-3,0.12),circle(3,-3,0.14),
circle(4,0,0.08),circle(4,0,0.10),circle(4,0,0.12),circle(4,0,0.14),
circle(5,5,0.08),circle(5,5,0.10),circle(5,5,0.12),circle(5,5,0.14),
circle(6,12,0.08),circle(6,12,0.10),circle(6,12,0.12),circle(6,12,0.14)

)}}}


<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=x^2-4x}}}:



{{{ drawing(500, 500, -10, 10, -5, 15,
grid(1),
graph(500, 500, -10, 10, -5, 15, x^2-4x),
circle(-2,12,0.08),circle(-2,12,0.10),circle(-2,12,0.12),circle(-2,12,0.14),
circle(-1,5,0.08),circle(-1,5,0.10),circle(-1,5,0.12),circle(-1,5,0.14),
circle(0,0,0.08),circle(0,0,0.10),circle(0,0,0.12),circle(0,0,0.14),
circle(1,-3,0.08),circle(1,-3,0.10),circle(1,-3,0.12),circle(1,-3,0.14),
circle(2,-4,0.08),circle(2,-4,0.10),circle(2,-4,0.12),circle(2,-4,0.14),
circle(3,-3,0.08),circle(3,-3,0.10),circle(3,-3,0.12),circle(3,-3,0.14),
circle(4,0,0.08),circle(4,0,0.10),circle(4,0,0.12),circle(4,0,0.14),
circle(5,5,0.08),circle(5,5,0.10),circle(5,5,0.12),circle(5,5,0.14),
circle(6,12,0.08),circle(6,12,0.10),circle(6,12,0.12),circle(6,12,0.14)

)}}} Graph of {{{y=x^2-4x}}}



So the graph of {{{y=x^2-4x}}} is


{{{ drawing(500, 500, -10, 10, -5, 15,
grid(1),
graph(500, 500, -10, 10, -5, 15, x^2-4x)
)}}}



Now shade the graph green from x=2 to the right like so:



{{{ drawing(500, 500, -10, 10, -5, 15,
grid(1),
graph(500, 500, -10, 10, -5, 15, x^2-4x,(sqrt(x-2)/sqrt(x-2))*x^2-4x)
)}}}



To find the inverse, we can do the following...



{{{y=x^2-4x}}} Start with the given equation.



{{{x=y^2-4y}}} Swap x and y



{{{x=y^2-4y+4-4}}} Take half of the y coefficient -4 to get -2. Square -2 to get 4. Add AND subtract this value on the right side.



{{{x=(y^2-4y+4)-4}}} Group the first three terms.



{{{x=(y-2)^2-4}}} Factor that group



{{{x+4=(y-2)^2}}} Add 4 to both sides.



{{{(y-2)^2=x+4}}} Rearrange the terms.



{{{y-2=""+-sqrt(x+4)}}} Take the square root of both sides.



{{{y=2+-sqrt(x+4)}}} Add 2 to both sides.



{{{y=2+sqrt(x+4)}}} or {{{y=2-sqrt(x+4)}}} Break up the 'plus/minus'



Now because the directions state that the graph has "the limited domain x>=2", this means that we're only going to focus on the positive portion of the inverse. So we're going to ignore {{{y=2-sqrt(x+4)}}}.



So the inverse of {{{y=x^2-4x}}} (over the domain *[Tex \LARGE x\ge 2]) is {{{y=2+sqrt(x+4)}}}



I'll let you finish up and graph the inverse.