Question

4. Identify the transformation as a reflection about the x-axis or a reflection about the y-axis of f(x) = x^4 - 3x by observing the equation of the function g(x) = -x^4 + 3x.

Answer 1

The transformation rules for reflection are given to be:

$\begin{gathered} f(x)\to-f(x)\text{ (}reflection\text{ over x-axis)} \\ f(x)\to f(-x)\text{ (}reflection\text{ over y-axis)} \end{gathered}$

The parent function is given to be:

$g(x)=-x^4+3x^{}$

The reflected function is:

$f(x)=x^4-3x$

Let us attempt to rewrite the reflected function such that it resembles the parent one:

$\begin{gathered} \text{Multiplying through the function by -1:} \\ -f(x)=-(x^4-3x) \\ -f(x)=-x^4+3x \end{gathered}$

Going by this, we can see that:

$g(x)=-f(x)$

Therefore, the reflection is about the x-axis.