To draw the graph of the form: 

y = a·|bx + c| + d

This is a graph shaped like this "⋁" or like this "⋀".

1.  Find the vertex (The sharp point of the graph) 
    A.  Set bx + c = 0 and solve for x.  This is the x-coordinate of the
        vertex.
    B.  The y-coordinate of the vertex is d.
    C.  Plot this point.

2.  Find a point on the left of the vertex
    A.  Select a value of x less than the x coordinate of the vertex
        for the x-ccordinate of a second point on the graph. 
    B.   Substitute this value into the equation to find the corresponding
        y-coordinate for that point.
    C.  Plot that point.

3.  Find a point on the right of the vertex
    A.  Select a value of x greater than the x coordinate of the vertex
        for the x-ccordinate of a third point on the graph. 
    B.  Substitute this value into the equation to find the corresponding
        y-coordinate for that point.
    C.  Plot that point.

4. Draw a graph through those three points shaped like this "⋁" or like
   this "⋀", with the vertex at the sharp point.

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Let's go through your problem with those rules.

y =   |3x - 6| + 1

Compare to

y = a·|bx + c| + d

a=1, b=3, c=-6, d=1 


1.  Find the vertex (The sharp point of the graph) 
    A.  Set bx + c = 0 and solve for x.  
            3x - 6 = 0 
                3x = 6
                 x = 2

    This is the x-coordinate of the vertex.
    B.  The y-coordinate of the vertex is d=1.
    C.  Plot this point, which is (2,1)

 

2.  Find a point on the left of the vertex
    A.  Select a value of x less than the x coordinate of the vertex
        for the x-ccordinate of a second point on the graph. 
     
    Let's select 1 which is less than 2 

    B.   Substitute this value into the equation to find the corresponding
         y-coordinate for that point.
      
         y = |3x - 6| + 1
         y = |3(1) - 6| + 1 
         y = |3 - 6| + 1
         y = |-3| + 1
         y = 3 + 1
         y = 4, so the second point is (1,4)

    C.  Plot that point.



3.  Find a point on the right of the vertex
    A.  Select a value of x greater than the x coordinate of the vertex
        for the x-ccordinate of a third point on the graph. 
     
    Let's select 3 which is greater than 2 

    B.   Substitute this value into the equation to find the corresponding
         y-coordinate for that point.
      
         y = |3x - 6| + 1
         y = |3(3) - 6| + 1 
         y = |9 - 6| + 1
         y = |3| + 1
         y = 3 + 1
         y = 4, so the third point is (3,4)

    C.  Plot that point.



4. Draw a graph through those three points shaped like this "⋁" or like
   this "⋀", with the vertex at the sharp point.

   In this case it is shaped like this "⋁", so we draw in the graph: 
   



Edwin