SOLUTION: Identify the maximum or minimum (vertex), zeros, and axis of symmetry line and show the graph. y=x^2-4x+1

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Question 413886: Identify the maximum or minimum (vertex), zeros, and axis of symmetry line and show the graph.
y=x^2-4x+1
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

The maximum or minimum point (vertex) of the graph of 

y+=+ax%5E2%2Bbx%2Bc

has x-coordinate = -b%2F%282a%29

If a is a positive number, the vertex is a minimum.
If a is a negative number, the vertex is a maximum 

Its y-coordinate is found by substituting the value of the x
coordinate for x into y+=+ax%5E2%2Bbx%2Bc

It's zeros are 

%28-b+%2B+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+,0) 

and

%28-b+%2B+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+,0)

The axis of symmetry is the vertical line which passes through the 
vertex and has the equation

x+=+-b%2F%282a%29

The y-intercept is (0,c)

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Identify the maximum or minimum (vertex), zeros, and axis of symmetry line and show the graph.

y=x^2-4x+1

a=1, b=-4, c=1


The maximum or minimum point (vertex) of the graph of 

y+=+ax%5E2%2Bbx%2Bc

y+=+1x%5E2-4x%2B1

a=1, b=-4, c=1

has x-coordinate = -b%2F%282a%29

                 = -%28-4%29%2F%282%281%29%29
  
                 = 4%2F2

                 = 2

a=1 is a positive number, so the vertex is a minimum.

Its y-coordinate is found by substituting the value of the x
coordinate for x into y+=+ax%5E2%2Bbx%2Bc

Substituting 2 for x

y+=+1x%5E2-4x%2B1
y+=+1%282%29%5E2-4%282%29%2B1
y+=+4-8%2B1
y+=+-3

So the y-coordinate is -3 and so the vertex has
coordinates:

(2, -3)

It's zeros are 


(%28-b+%2B+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+,0) 

and

(%28-b+%2B+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+,0)

Substituting  a=1, b=-4, c=1


(%28-%28-4%29+%2B+sqrt%28%28-4%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29+,0)

(%284+%2B+sqrt%2816-4%29%29%2F2+,0)

(%284+%2B+sqrt%2812%29%29%2F2+,0)

(%284+%2B+sqrt%284%2A3%29%29%2F2+,0)

(%284+%2B+2sqrt%283%29%29%2F2+,0)

(%282%282+%2B+sqrt%283%29%29%29%2F2+,0)

(%28cross%282%29%282+%2B+sqrt%283%29%29%29%2Fcross%282%29+,0)

(2+%2B+sqrt%283%29+,0)

Similarly just be changing the sign before the radical term
above, the other zero is

(2+-+sqrt%283%29+,0)

They are approximately (.27,0) and (3.73,0)

The axis of symmetry is the vertical line which passes through the 
vertex and has the equation

x+=+-b%2F%282a%29

which is

x+=+2

We plot the vertex, the x-intercepts and the axis of symmetry (in green).
and the y-intercept is (0,c)



Then sketch in the graph:




Edwin