SOLUTION: Can you please show me how to graph each quadratic equation using its properties, and also based on the graph, determine the domain and range of the quadratic function. I am los

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Can you please show me how to graph each quadratic equation using its properties, and also based on the graph, determine the domain and range of the quadratic function. I am los      Log On


   

Question 171579: Can you please show me how to graph each quadratic equation using its properties, and also based on the graph, determine the domain and range of the quadratic function.
I am lost when it comes to this new chapter we have started...
F(x) = -x^2 + 2x + 8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
In order to graph f%28x%29=-x%5E2%2B2x%2B8, 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)

Finding the vertex:




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=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=-x%5E2%2B2x%2B8, we can see that a=-1, b=2, and c=8.


x=%28-%282%29%29%2F%282%28-1%29%29 Plug in a=-1 and b=2.


x=%28-2%29%2F%28-2%29 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%5E2%2B2x%2B8 Start with the given equation.


y=-%281%29%5E2%2B2%281%29%2B8 Plug in x=1.


y=-1%281%29%2B2%281%29%2B8 Square 1 to get 1.


y=-1%2B2%281%29%2B8 Multiply -1 and 1 to get -1.


y=-1%2B2%2B8 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 .


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Step 2)

Find two points to the left of the axis of symmetry:




Lets find the value of y when x=-1


f%28x%29=-x%5E2%2B2x%2B8 Start with the given polynomial


f%28-1%29=-%28-1%29%5E2%2B2%28-1%29%2B8 Plug in x=-1


f%28-1%29=-%281%29%2B2%28-1%29%2B8 Square 1 to get 1


f%28-1%29=-%281%29-2%2B8 Multiply 2 by -1 to get -2


f%28-1%29=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%28x%29=-x%5E2%2B2x%2B8 Start with the given polynomial


f%280%29=-%280%29%5E2%2B2%280%29%2B8 Plug in x=0


f%280%29=0%2B2%280%29%2B8 Square 0 to get 0


f%280%29=0%2B0%2B8 Multiply 2 by 0 to get 0


f%280%29=8 Combine terms


So our 2nd point is (0,8)


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Step 3)

Reflecting the two points over the axis of symmetry:




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)


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Step 4)

Plotting the points:




Now lets make a table of the values we have calculated 



 xy


 -15
 

 08
 

 19
 

 28
 

 35



Now let's plot the points:





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


Step 5)

Drawing a curve through all of the points:




Now draw a curve through all of the points to graph y=-x%5E2%2B2x%2B8:


Graph of y=-x%5E2%2B2x%2B8




From the graph, we can determine the following:


Domain: All real numbers

Range: y%3C=9