Respuesta :
The transformed function is given as,
[tex]f(x)=-2(x-3)^4+1[/tex]The function is a transformation of the parent function,
[tex]g(x)=x^4[/tex]Consider that the horizontal translation by right by 'c' units is given by,
[tex]g(x)\rightarrow g(x-c)[/tex]Applying the definition, it seems that the first transformation is a translation of the function right by 3 units.
After this first transformation, the function will become,
[tex]g(x)=(x-3)^4[/tex]Now, consider the next transformation, that is, vertical stretch (|a|>0),
[tex]g(x)\rightarrow a\cdot g(x)[/tex]Applying the definition to the function,
[tex](x-3)^4\rightarrow2(x-3)^4[/tex]Thus, the second transformation will be the vertical stretch.
Now, consider the reflection of the function about the x-axis, is characterized by,
[tex]g(x)\rightarrow-g(x)[/tex]Applying the definition to the definition,
[tex]2(x-3)^4\rightarrow-2(x-3)^4[/tex]Thus, the third transformation will be a reflection about the x-axis.
Now, consider that the transformation of vertical translation up by 'd' units, is characterized as,
[tex]g(x)\rightarrow g(x)+d[/tex]Applying the definition,
[tex]-2(x-3)^4\rightarrow-2(x-3)^4+1[/tex]This represents the vertical translation of the function by 1 unit.
Thus, the fourth transformation will be the vertical translation by 1 unit.
And finally, the transformed function is obtained.
Thus, it can be concluded that the parent function undergoes the following transformation sequence to be the given function,
1. Translation right by 3 units.
2. Vertical stretch by a factor of 2.
3. Reflection about the x-axis.
4. Vertical translation by 1 unit.
