Question

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 1

`y=-x^(2)+2x`

`y=-x^(2)+2x-4`

`y=-x^(2)+2x-3`

Explanation:

we know that

The equation of a vertical parabola in vertex form is equal to

`y=a(x-h)^(2) +k`

where

(h,k) is the vertex

The axis of symmetry is equal to the x-coordinate of the vertex

so

`x=h`

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

`y=-x^(2) +2x+3`

The vertex is the point
`(1,4)` ------> observing the graph

The axis of symmetry is
`x=1`

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)
`y=-x^(2)+2x`

Convert to vertex form

`y=-(x^(2)-2x)`

`y-1=-(x^(2)-2x+1)`

`y-1=-(x-1)^(2)`

`y=-(x-1)^(2)+1`

The vertex is (1,1)

therefore

The function could be the equation of the new graph

case b)
`y=-x^(2)-2x+3`

Convert to vertex form

`y-3=-(x^(2)+2x)`

`y-3-1=-(x^(2)+2x+1)`

`y-4=-(x+1)^(2)`

`y=-(x+1)^(2)+4`

The vertex is (-1,4)

therefore

The function cannot be the equation of the new graph

case c)
`y=-x^(2)+2x-4`

Convert to vertex form

`y+4=-(x^(2)-2x)`

`y+4-1=-(x^(2)-2x+1)`

`y+3=-(x-1)^(2)`

`y=-(x-1)^(2)-3`

The vertex is (1,-3)

therefore

The function could be the equation of the new graph

case d)
`y=-x^(2)+2x+4`

Convert to vertex form

`y-4=-(x^(2)-2x)`

`y-4-1=-(x^(2)-2x+1)`

`y-5=-(x-1)^(2)`

`y=-(x-1)^(2)+5`

The vertex is (1,5)

therefore

The function cannot be the equation of the new graph

case e)
`y=-x^(2)+2x-3`

Convert to vertex form

`y+3=-(x^(2)-2x)`

`y+3-1=-(x^(2)-2x+1)`

`y+2=-(x-1)^(2)`

`y=-(x-1)^(2)-2`

The vertex is (1,-2)

therefore

The function could be the equation of the new graph