SOLUTION: Graph and answer the questions. f(x) = -x^2 + 4x - 1 What are the coordinates of the vertex? What is the domain? What is the range?

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Question 83862: Graph and answer the questions.
f(x) = -x^2 + 4x - 1
What are the coordinates of the vertex?
What is the domain?
What is the range?
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
In order to graph -x%5E2%2B4x-1, we need to plot some points.
So we can start at any x value. So lets start at x=-1

y=-x%5E2%2B4x-1 Start with the given equation

y=-%28-1%29%5E2%2B4%28-1%29-1 Plug in x=-1

-%28-1%29%5E2%2B4%28-1%29-1=-6 Calculate the y value
So when x=-1 y=-6
So our 1st point is (-1,-6)




y=-x%5E2%2B4x-1 Start with the given equation

y=-%280%29%5E2%2B4%280%29-1 Plug in x=0

-%280%29%5E2%2B4%280%29-1=-1 Calculate the y value
So when x=0 y=-1
So our 2nd point is (0,-1)




y=-x%5E2%2B4x-1 Start with the given equation

y=-%281%29%5E2%2B4%281%29-1 Plug in x=1

-%281%29%5E2%2B4%281%29-1=2 Calculate the y value
So when x=1 y=2
So our 3rd point is (1,2)




y=-x%5E2%2B4x-1 Start with the given equation

y=-%282%29%5E2%2B4%282%29-1 Plug in x=2

-%282%29%5E2%2B4%282%29-1=3 Calculate the y value
So when x=2 y=3
So our 4th point is (2,3)




y=-x%5E2%2B4x-1 Start with the given equation

y=-%283%29%5E2%2B4%283%29-1 Plug in x=3

-%283%29%5E2%2B4%283%29-1=2 Calculate the y value
So when x=3 y=2
So our 5th point is (3,2)




y=-x%5E2%2B4x-1 Start with the given equation

y=-%284%29%5E2%2B4%284%29-1 Plug in x=4

-%284%29%5E2%2B4%284%29-1=-1 Calculate the y value
So when x=4 y=-1
So our 6th point is (4,-1)




y=-x%5E2%2B4x-1 Start with the given equation

y=-%285%29%5E2%2B4%285%29-1 Plug in x=5

-%285%29%5E2%2B4%285%29-1=-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 



 xy



 -1-6
 

 0-1
 

 12
 

 23
 

 32
 

 4-1
 

 5-6
Now plot the points
drawing%28+900%2C+900%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++grid%28+1+%29%2Ccircle%28-1%2C-6%2C0.05%29%2C+%0D%0A++circle%28-1%2C-6%2C0.08%29%2C+%0D%0Acircle%280%2C-1%2C0.05%29%2C+%0D%0A++circle%280%2C-1%2C0.08%29%2C+%0D%0Acircle%281%2C2%2C0.05%29%2C+%0D%0A++circle%281%2C2%2C0.08%29%2C+%0D%0Acircle%282%2C3%2C0.05%29%2C+%0D%0A++circle%282%2C3%2C0.08%29%2C+%0D%0Acircle%283%2C2%2C0.05%29%2C+%0D%0A++circle%283%2C2%2C0.08%29%2C+%0D%0Acircle%284%2C-1%2C0.05%29%2C+%0D%0A++circle%284%2C-1%2C0.08%29%2C+%0D%0Acircle%285%2C-6%2C0.05%29%2C+%0D%0A++circle%285%2C-6%2C0.08%29+%0D%0A%29
Now connect the points, this is the graph of -x%5E2%2B4x-1
drawing%28+900%2C+900%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++grid%28+1+%29%2C%0D%0A++graph%28+900%2C+900%2C-10%2C+10%2C+-10%2C+10%2C+-x%5E2%2B4x-1%29%2Ccircle%28-1%2C-6%2C0.05%29%2C+%0D%0A++circle%28-1%2C-6%2C0.08%29%2C+%0D%0Acircle%280%2C-1%2C0.05%29%2C+%0D%0A++circle%280%2C-1%2C0.08%29%2C+%0D%0Acircle%281%2C2%2C0.05%29%2C+%0D%0A++circle%281%2C2%2C0.08%29%2C+%0D%0Acircle%282%2C3%2C0.05%29%2C+%0D%0A++circle%282%2C3%2C0.08%29%2C+%0D%0Acircle%283%2C2%2C0.05%29%2C+%0D%0A++circle%283%2C2%2C0.08%29%2C+%0D%0Acircle%284%2C-1%2C0.05%29%2C+%0D%0A++circle%284%2C-1%2C0.08%29%2C+%0D%0Acircle%285%2C-6%2C0.05%29%2C+%0D%0A++circle%285%2C-6%2C0.08%29+%0D%0A%29

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%3C=3