Question

identify the transformation of the give equations below compared to its parent
`y = - 2(x - 5)^(2) + 1`a)Vertical Stretch by a factor of 2, shifts left 5units, shifts down 1 unit and reflection .b)Vertical compression by a factor of 2, shifts right 5 units, shifts up 1 unit and reflection across the x-axis .c)Vertical stretch by a factor of 2, shifts right units, shifts up 1 unit and reflection across the x-axis .d)Vertical compression by a factor of 2, shifts left 5 units, shifts down 1 unit and reflection across the x-axis .

Answer 1

Let's begin by identifying key information given to us:

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

The general formula for the transformation is:

`\begin{gathered} f\mleft(x\mright)=a\mleft(x-h\mright)^2+k \\ where\colon \\ h=HorizontalShift, \\ k=VerticalShift \\ a=Stretched/Compressed \end{gathered}`

a)

Vertical Stretch by a factor of 2, shifts left 5units, shifts down 1 unit, and reflection gives:

`\begin{gathered} a=2,h=-5,k=-1 \\ Reflection=-a \\ \Rightarrow g(x)=-2(x+5)^2-1 \end{gathered}`

b)

Vertical compression by a factor of 2, shifts right 5 units, shifts up 1 unit, and reflection across the x-axis gives:

`\begin{gathered} a=(1)/(2),h=-5,k=1 \\ Reflection=-a \\ \Rightarrow g(x)=-(1)/(2)(x+5)+1 \end{gathered}`

c)

Vertical stretch by a factor of 2, shifts right units, shifts up 1 unit, and reflection across the x-axis gives:

`\begin{gathered} a=2,h=5,k=1 \\ Reflection=-a \\ \Rightarrow g(x)=-2(x-5)^2+1 \end{gathered}`