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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?

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    You are right and your teacher is wrong. – 2017-02-16
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    Nope, you should be right. If they continue to disagree, make a few graphs (one with your choice and one with his/her choice of answer) and ask which is right. Then tell her which graph was which and see what (s)he thinks. – 2017-02-16
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    Thanks guys! I made a few graphs using Desmos and even made a table of values for myself, I will talk to him about it tomorrow. – 2017-02-16

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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.

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    So if I take the function f(x) = |x| + 1 and I reflect f(x) given its definition, it is not -f(x)? Sorry, I am a little confused as I was taught that vertical reflections are always symmetric about the x-axis. In this case, if f(x) = |x| + 1 and the reflected function is g(x) = -|x| + 1 then wouldn't that be a reflection about the line y = 1 and not the x-axis? – 2017-02-17
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    @ZacheryKish The equation you came up with is symmetric about the x axis.. i.e for x the result is -f(x)... a reflection is taking the graph and flipping it upside down directly in its position, refer back to the vertexes i mentioned. Use a graphing calculator to graph all three functions and you will get a visual of what is going on. If you dont have such calculator try wolfram alpha. – 2017-02-17
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    Ok thanks for your hasty responses but I am still super confused. I'm looking at my notes right now, and the way my teacher defined a vertical reflection is as follows, "y = f(x) undergoes a vertical reflection in the x-axis and becomes y = -f(x)." So I understand that my equation is symmetric and not a reflection but do you see where my confusion is coming from? – 2017-02-17
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    @ZacheryKish I think the graphs are easier to understand. Reflection maintains the same vertex while symmetry is y=-f(x). But there are cases where the overlab. F(x) = x and G(x) = -x are reflections as well as symmetric. – 2017-02-17
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    Ok so basically when looking at a function such as f(x) = |x| + 1 and when I am told that there is a vertical reflection, the reflection is based on the base function, g(x) = |x| and not the whole function f(x) = |x| + 1 because then that would be looking at vertical symmetry in the x-axis and not reflection, correct? I have graphed both f(x) = |x|, g(x) = -|x| - 1 and h(x) = -|x| + 1 and I understand what you are talking about. Basically the reflection occurs based on the base function first and then the translation of one unit up to obtain g(x) = -|x| + 1. – 2017-02-17
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    @ZacheryKish Exactly! Good stuff! It sometimes can be a little tricky but those are the suttle differences. – 2017-02-17
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    @ZacheryKish The best way to tell is to check that the vertexes are the same – 2017-02-17
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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. :)

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    Okay thanks for your response, however, how do I know for future reference what is correct and which way to go about solving problems like this? I understand the solution in both ways, but I do not know which one I should use in future situations? Does this mainly depend on the given question? – 2017-02-17
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    Absolutely. The language in the question should be improved. It's on your teachers end I think. Saying the absolute value of x is the base function without any prior knowledge isn't fair. You were asked to reflect the function $f(x)$ and you did. So it all depends what your teacher says as an excuse. – 2017-02-17