The graph of this function is shifted downwards and the axis of symmetry remains x=1. Which function below the equation of the new graph select all correct answer

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 thatThe equation of a vertical parabola in vertex form is equal to[tex]y=a(x-h)^{2} +k[/tex]where(h,k) is the vertexThe axis of symmetry is equal to the x-coordinate of the vertexso[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 graphThe axis of symmetry is [tex]x=1[/tex]If the graph of this function is shifted downwards and the axis of symmetry remains x=1then The x-coordinate of the vertex of the new graph must be equal to 1The y-coordinate of the vertex of the new graph must be less than 4The parabola of the new graph open downwardthereforeVerify each casecase 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)thereforeThe function could be the equation of the new graphcase 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)thereforeThe function cannot be the equation of the new graphcase 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)thereforeThe function could be the equation of the new graphcase 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)thereforeThe function cannot be the equation of the new graphcase 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)thereforeThe function could be the equation of the new graph