Function reflection question

Question

So I wrote an exam the other day for pre-calculus grade 12 and this was one of the questions ...

Given f(x) = |x| + 1, determine the equation of the tranformation function when f(x) is reflected in the x-axis.

So from my understanding, I expressed the transformation function in terms of g(x) where g(x) = -f(x) = -|x| - 1. However, my teacher said that "the transformation fucntion should be g(x) = -|x| + 1 because y = |x| is the base function in f(x)." Moreover, he did not help me to understand this any further, what am I misunderstanding?

Answer 1

given

f(x) = |x| + 1,

You have the base function f(x) = |x| shifted up 1 unit. Therefore the vertex of f(x) is (0,1)

Saying that

g(x) = - f(x) = -|x| - 1,

You have base function g(x) = -|x| shifted down one unit. So, the base function is reflected about the x axis, however; the vertex of g(x) is (0,-1).

Your functions have different vertexes, and are then not reflections of each other. That is why

g(x) = -|x| + 1

, reflecting the graph about the x axis, and maintaining the vertex of (0,1). The solution for which you provided is in regards to symmetry about the x axis, not reflection.

Answer 2

Well, $$f(x)=\left|x\right|+1$$ This then reflected in the x-axis will give a new function $$g(x)=-\left|x\right|-1.$$ You teacher must be referring to the ordering of transformation. If the original function was in turn transformed then the base function is $f(x)=\vert x\vert$ which is then translated by $+1$ in the y-axis.

This would then mean that a reflection of the base function would indeed be $$g(x)=-\left|x\right|$$ which would then be translated in the same way to yield $$g(x)=-\left|x\right|+1$$

But see what he explains tomorrow, don't leave the room without a proper explanation either. :)