Question 171579
In order to graph {{{f(x)=-x^2+2x+8}}}, we can follow the steps:



Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph)



Step 2) Once you have the vertex, find two points on the left side of the axis of symmetry (the line that vertically runs through the vertex)



Step 3) Reflect the points over the axis of symmetry to get two more points on the right side of the parabola (remember a parabola is symmetric).



Step 4) Plot all of the points found (including the vertex)



Step 5) Draw a curve through all of the points to graph the parabola



Let's go through these steps in detail



Step 1) <h3>Finding the vertex:</h3>



In order to find the vertex, we first need to find the x-coordinate of the vertex.



To find the x-coordinate of the vertex, use this formula: {{{x=(-b)/(2a)}}}.



{{{x=(-b)/(2a)}}} Start with the given formula.



From {{{y=-x^2+2x+8}}}, we can see that {{{a=-1}}}, {{{b=2}}}, and {{{c=8}}}.



{{{x=(-(2))/(2(-1))}}} Plug in {{{a=-1}}} and {{{b=2}}}.



{{{x=(-2)/(-2)}}} Multiply 2 and {{{-1}}} to get {{{-2}}}.



{{{x=1}}} Divide.



So the x-coordinate of the vertex is {{{x=1}}}. Note: this means that the axis of symmetry is also {{{x=1}}}.



Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.



{{{y=-x^2+2x+8}}} Start with the given equation.



{{{y=-(1)^2+2(1)+8}}} Plug in {{{x=1}}}.



{{{y=-1(1)+2(1)+8}}} Square {{{1}}} to get {{{1}}}.



{{{y=-1+2(1)+8}}} Multiply {{{-1}}} and {{{1}}} to get {{{-1}}}.



{{{y=-1+2+8}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}.



{{{y=9}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=9}}}.



So the vertex is *[Tex \LARGE \left(1,9\right)].



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



Step 2) <h3>Find two points to the left of the axis of symmetry:</h3>



Lets find the value of y when x=-1



{{{f(x)=-x^2+2x+8}}} Start with the given polynomial



{{{f(-1)=-(-1)^2+2(-1)+8}}} Plug in {{{x=-1}}}



{{{f(-1)=-(1)+2(-1)+8}}} Square 1 to get 1



{{{f(-1)=-(1)-2+8}}} Multiply 2 by -1 to get -2



{{{f(-1)=5}}} Now combine like terms



So our 1st point is (-1,5)




----Now lets find another point----




Lets find the y value when x=0


{{{f(x)=-x^2+2x+8}}} Start with the given polynomial



{{{f(0)=-(0)^2+2(0)+8}}} Plug in {{{x=0}}}



{{{f(0)=0+2(0)+8}}} Square 0 to get 0



{{{f(0)=0+0+8}}} Multiply 2 by 0 to get 0



{{{f(0)=8}}} Combine terms



So our 2nd point is (0,8)



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



Step 3) <h3>Reflecting the two points over the axis of symmetry:</h3>



Now remember, the parabola is symmetrical about the axis of symmetry (which is {{{x=1}}})



This means the y-value for {{{x=0}}} is equal to the y-value of {{{x=2}}}. So when {{{x=2}}}, {{{y=8}}}. So we've reflected the point (0,8) over to (2,8)



Also, the y-value for {{{x=-1}}} is equal to the y-value of {{{x=3}}}. So when {{{x=3}}}, {{{y=5}}}. So we've reflected the point (-1,5) over to (3,5)



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



Step 4) <h3>Plotting the points:</h3>



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>5</TD></TR> 
<TR><TD> 0</TD><TD>8</TD></TR> 
<TR><TD> 1</TD><TD>9</TD></TR> 
<TR><TD> 2</TD><TD>8</TD></TR> 
<TR><TD> 3</TD><TD>5</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(-1,5,0.08),circle(-1,5,0.10),
circle(0,8,0.08),circle(0,8,0.10),
circle(1,9,0.08),circle(1,9,0.10),
circle(2,8,0.08),circle(2,8,0.10),
circle(3,5,0.08),circle(3,5,0.10)

)}}}



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



Step 5) <h3>Drawing a curve through all of the points:</h3>



Now draw a curve through all of the points to graph {{{y=-x^2+2x+8}}}:



{{{ drawing(500, 500, -10, 10, -5, 15,
grid(1),
graph(500, 500, -10, 10, -5, 15, -x^2+2x+8),
circle(-1,5,0.08),circle(-1,5,0.10),
circle(0,8,0.08),circle(0,8,0.10),
circle(1,9,0.08),circle(1,9,0.10),
circle(2,8,0.08),circle(2,8,0.10),
circle(3,5,0.08),circle(3,5,0.10)

)}}} Graph of {{{y=-x^2+2x+8}}}





From the graph, we can determine the following:



Domain: All real numbers


Range: {{{y<=9}}}