AI vs Textbook: Reflecting a vertically shifted function should adhere to x = 0 axis or x = added distance?

Question

I'm relearning algebra by taking a course online, starting from the very basic now on the topic of "Shifts and Reflections of Functions" and kind of have an uncertain grasp of how the "textbook" explains an example, which is stated as

With the parent function f(x) = x², shift the function down by seven and three to the right. Then, reflect it over the x-axis.

The explanation goes to:

1. start with f(x) = x²

2. Shift down 7 so that g(x) = x² - 7.

3. Then shift right 3 so that g(x) = (x-3)² - 7.

things are going as I expected until the next step

4. Finally, add the vertical reflection so that g(x) =-(x-3)² - 7. Since the entire original function has already moved 7 below the x-axis at this point, it will actually be reflected over the line y = -7, instead of y = 0 (the x-axis).

5. In the end, g(x) will resemble the following graph.

Now, what I know of reflection is, in the case of vertically reflecting, reflecting over x-axis with the x number remaining the same and changing the sign of the y number, which is essentially multiplied -1, I think.

But the explanation from the textbook here also makes sense. So with this confusion, I fed the question to Claude.ai, which disagreed with what the textbook said and responded:

No, I disagree with that explanation, and here's why:

When we reflect over the x-axis (y = 0), we multiply the ENTIRE function by -1, which means we multiply everything in the function by -1. It doesn't matter where the function is located when we do the reflection - the x-axis (y = 0) is always our line of reflection.

Let's see what happens step by step:

1. Start with (x-3)² - 7
2. Multiply EVERYTHING by -1: * -1[(x-3)² - 7] * = -(x-3)² + 7

It's like looking at the function in a mirror placed along y = 0. Everything that was below the x-axis goes above it by the same distance, and vice versa.

The explanation you saw suggesting "it will actually be reflected over the line y = -7" is incorrect because:

1. Reflection over the x-axis is ALWAYS over y = 0, not any other horizontal line
2. The -7 is part of the function that gets multiplied by -1 during reflection
3. If we reflected over y = -7 as they suggest, we'd get a completely different graph

Think of it this way: if you have a point at (3,-7) and reflect it over y = 0, it goes to (3,7). That's what we want! But if we reflected over y = -7, that point would only move to (3,-7), which isn't a reflection at all.

So the answer should be f(x) = -(x-3)² + 7.

Now, it's more confusing... because I find Claude.ai's explanation also makes sense and kind of is what I expected before I read the textbook's explanation, but I'm also aware how AI can be so confidently misleading... and the explanation from ChatGPT confuses you even more because it just changes its stand back and forth like a politician.....

In a word, after the function shifted vertically to this, g(x) = (x-3)² - 7, when it performs reflection over the x-axis, should it be g(x) = - (x-3)² - 7 or g(x) = - (x-3)² + 7?

Thanks!

ps: I had checked similar posts like this but I find no inspiration from there, hopefully I can get some insights from opening a new one.

Answer 1

I think $g(x)=-(x-3)^2+7$ would be the final answer.

Here's why: In the $3^\mathrm{rd}$ step, you reached the function $(x-3)^2-7$. Now, to reflect it about $x-$axis, or as you said, multiply it be $-1$, that $-1$ will be multiplied to the whole function and not only the squared term. To reflect the graph of a function about $x-$axis means that all the values of the function would be multipled by $-1$.

So, $g(x)=-(x-3)^2+7$ would be the final answer after transformations.