https://file.aiquickdraw.com/mathbot/file/1727283621989kge4a8jl.jpg

In the image, we have a graph with two curves: a black one labeled as \( y = f(x) \) and a red one for which the equation is to be determined. The instructions mention the "big idea of opposites," which suggests that this question is about transformations of the function \( f(x) \).

Looking at the options and the visual clues:

1. The red graph appears to be a reflection of the black graph across the y-axis. This is a typical transformation that occurs when we replace \( x \) with \( -x \) in the function's argument, i.e., \( y = f(-x) \).

Now, let's match this concept with the choices:

- **Option A: \( y = f(x/(-1)) \)** — This is equivalent to \( y = f(-x) \), which fits the description of a reflection across the y-axis.
- **Option B: \( y - 1 = f(x) \)** — This describes a vertical shift down by 1 unit, which doesn't match the reflection.
- **Option C: \( y = f(x - 1) \)** — This describes a horizontal shift to the right by 1 unit, not a reflection.
- **Option D: \( y(-1) = f(x) \)** — This is unclear and does not match any standard transformation notation.

Thus, **Option A: \( y = f(x/(-1)) \)** is the correct equation for the red graph, representing the reflection of \( f(x) \) across the y-axis. 

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How does changing \( f(x) \) to \( f(-x) \) affect the graph?
2. What is the difference between reflecting across the y-axis and reflecting across the x-axis?
3. What happens to the graph when you apply a vertical or horizontal shift?
4. Can you explain the effect of stretching or compressing a graph using scaling transformations?
5. What transformation is represented by \( y = -f(x) \)?

**Tip:** Always consider how modifying \( x \) or \( y \) inside or outside the function affects the graph's shape or position.

The black graph is y = f(x). Choose the equation for the red graph. The options are:
A. y = f(x/(-1))
B. y - 1 = f(x)
C. y = f(x - 1)
D. y(-1) = f(x)

Learn how to determine the equation of a reflected graph using function transformations. This problem demonstrates the concept of reflecting a graph across the y-axis by applying the transformation y = f(-x). Suitable for students in Grades 9-11.

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