Question

List the sequence of transformations. You must use the sequence from our notes.()=−2(−3)^4+1

Answer 1

The transformed function is given as,

`f(x)=-2(x-3)^4+1`

The function is a transformation of the parent function,

`g(x)=x^4`

Consider that the horizontal translation by right by 'c' units is given by,

`g(x)\rightarrow g(x-c)`

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,

`g(x)=(x-3)^4`

Now, consider the next transformation, that is, vertical stretch (|a|>0),

`g(x)\rightarrow a\cdot g(x)`

Applying the definition to the function,

`(x-3)^4\rightarrow2(x-3)^4`

Thus, the second transformation will be the vertical stretch.

Now, consider the reflection of the function about the x-axis, is characterized by,

`g(x)\rightarrow-g(x)`

Applying the definition to the definition,

`2(x-3)^4\rightarrow-2(x-3)^4`

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,

`g(x)\rightarrow g(x)+d`

Applying the definition,

`-2(x-3)^4\rightarrow-2(x-3)^4+1`

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.