(-1)^floor(x)*(f1(mod(x,1))-.5)+.5 MAKES MIRORED ON X AXIS AT POINT Y 0.5 COPYS OF F1(x) WITHIN X AXIS . 

IS ANY FUNCTION THAT GIVES MIRORED ON Y AXIS AT POINT X 0.5 COPYS OF F1(x) WITHIN X AXIS 

IN Y=F(x) PLOTER

To create a function that mirrors copies of \( f_1(x) \) across the y-axis at the point \( x = 0.5 \) within the x-axis, you can construct a function similar to the one you provided for the x-axis mirroring.

### Explanation

1. **Original Function:** 
   The function you provided:
   \[
   (-1)^{\lfloor x \rfloor} \left( f_1(\text{mod}(x, 1)) - 0.5 \right) + 0.5
   \]
   mirrors the function \( f_1(x) \) on the x-axis at the point \( y = 0.5 \) repeatedly for integer intervals.

2. **Y-Axis Mirroring:**
   To mirror a function \( f_1(x) \) across the y-axis at \( x = 0.5 \), the transformation you need would involve reflecting the function around \( x = 0.5 \). For this, consider the transformation \( 1 - x \), but offset it by \( 0.5 \) to shift the mirroring axis.

   The mirrored function would be:
   \[
   f(x) = f_1\left(1 - \text{mod}(x - 0.5, 1)\right)
   \]
   This creates mirrored copies of \( f_1(x) \) centered at \( x = 0.5 \).

3. **Combining Copies:**
   To repeatedly copy and mirror \( f_1(x) \) within intervals along the x-axis, combine this mirroring transformation with integer shifts:
   \[
   f(x) = f_1\left(\text{mod}(x, 1)\right) + f_1\left(1 - \text{mod}(x, 1)\right) - f_1(0.5)
   \]
   Here, \( f_1(0.5) \) is subtracted to center the function correctly.

### Summary
The function to create mirrored copies of \( f_1(x) \) across the y-axis at \( x = 0.5 \) is:
\[
f(x) = f_1\left(1 - \text{mod}(x - 0.5, 1)\right)
\]
This will give you the desired mirroring effect centered on \( x = 0.5 \).

Would you like more details on the function construction or have any specific questions?

### Follow-up Questions

1. How does the mod function influence the pattern of the graph?
2. What is the significance of the floor function in mirroring across axes?
3. Can you explain why \( x = 0.5 \) is chosen as the mirroring point?
4. How would the behavior of \( f_1(x) \) change with different mod intervals?
5. What happens if we change the mirroring point from \( x = 0.5 \) to another value?
6. Can the function be generalized for mirroring across other points?
7. How would you graphically represent the mirrored function on a plotting tool?
8. How does this function behave with non-continuous or non-smooth functions?

### Tip:
When working with transformations like these, visualizing the function graphically helps to ensure the transformations are implemented correctly.

Explore how to create mirrored copies of f1(x) across the y-axis at x = 0.5 using mathematical transformations. Understand the function f(x) = f_1(1 - mod(x - 0.5, 1)) and its applications in graphical plotting.

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