How Does One Write the Equation of a Transformed Function, Given the Transformation

Question

After using my math textbook, Desmos and Photomath to determine the answer, I am still confused as to what the process is.

E.g.

If f(x) = x^4, determine the equation of the transformed function when y = -2f(2x+6)-4 is applied as the transformations.

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My understanding is that the equation would be:

= -2[2(x+6)]^4 -4

... but I have been told this is wrong and that the +6 is incorrect.

What is the problem with my new equation? Is the +6 correct?

Answer 1

The input to $f$ is $2x+6$,

hence $$y = -2(2x+6)^4 - 4$$

We can also write it as

$$y = -2(2(x+\color{red}3))^4 - 4$$

since $2x+6=2(x+3)$.

Answer 2

After you transform the function you obtain: $y=-2(2x+6)^4 -4$, in your answer you are raising $2(x+6)$ to the power 4, i.e $(2x+12)^4.$ The rest is all correct though.