Answer:
[tex]y=-x^{2}+2x[/tex]
[tex]y=-x^{2}+2x-4[/tex]
[tex]y=-x^{2}+2x-3[/tex]
Step-by-step explanation:
we know that
The equation of a vertical parabola in vertex form is equal to
[tex]y=a(x-h)^{2} +k[/tex]
where
(h,k) is the vertex
The axis of symmetry is equal to the x-coordinate of the vertex
so
[tex]x=h[/tex]
If a> 0 then the parabola open upward (vertex is a minimum)
If a< 0 then the parabola open downward (vertex is a maximum)
In this problem we have
[tex]y=-x^{2} +2x+3[/tex]
The vertex is the point [tex](1,4)[/tex] ------> observing the graph
The axis of symmetry is [tex]x=1[/tex]
If the graph of this function is shifted downwards and the axis of symmetry remains x=1
then
The x-coordinate of the vertex of the new graph must be equal to 1
The y-coordinate of the vertex of the new graph must be less than 4
The parabola of the new graph open downward
therefore
Verify each case
case a) [tex]y=-x^{2}+2x[/tex]
Convert to vertex form
[tex]y=-(x^{2}-2x)[/tex]
[tex]y-1=-(x^{2}-2x+1)[/tex]
[tex]y-1=-(x-1)^{2}[/tex]
[tex]y=-(x-1)^{2}+1[/tex]
The vertex is (1,1)
therefore
The function could be the equation of the new graph
case b) [tex]y=-x^{2}-2x+3[/tex]
Convert to vertex form
[tex]y-3=-(x^{2}+2x)[/tex]
[tex]y-3-1=-(x^{2}+2x+1)[/tex]
[tex]y-4=-(x+1)^{2}[/tex]
[tex]y=-(x+1)^{2}+4[/tex]
The vertex is (-1,4)
therefore
The function cannot be the equation of the new graph
case c) [tex]y=-x^{2}+2x-4[/tex]
Convert to vertex form
[tex]y+4=-(x^{2}-2x)[/tex]
[tex]y+4-1=-(x^{2}-2x+1)[/tex]
[tex]y+3=-(x-1)^{2}[/tex]
[tex]y=-(x-1)^{2}-3[/tex]
The vertex is (1,-3)
therefore
The function could be the equation of the new graph
case d) [tex]y=-x^{2}+2x+4[/tex]
Convert to vertex form
[tex]y-4=-(x^{2}-2x)[/tex]
[tex]y-4-1=-(x^{2}-2x+1)[/tex]
[tex]y-5=-(x-1)^{2}[/tex]
[tex]y=-(x-1)^{2}+5[/tex]
The vertex is (1,5)
therefore
The function cannot be the equation of the new graph
case e) [tex]y=-x^{2}+2x-3[/tex]
Convert to vertex form
[tex]y+3=-(x^{2}-2x)[/tex]
[tex]y+3-1=-(x^{2}-2x+1)[/tex]
[tex]y+2=-(x-1)^{2}[/tex]
[tex]y=-(x-1)^{2}-2[/tex]
The vertex is (1,-2)
therefore
The function could be the equation of the new graph
