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Let's go through the problems one by one:

### Problem 5:
1. Given \( S = \{ (x, y) \mid y = x^2 \} \) and \( A(3,1) \).
   - **5a.** Determine if \( B(3, -7) \in S' = H_A(S) \).
   - **5b.** Write the equation for \( S' \).

**Solution for 5:**
- To find \( S' \), we need to perform a reflection of \( S \) across point \( A(3,1) \).
- The reflection of \( S \) across \( A \) would involve transforming each point \((x, y)\) on \( y = x^2 \) relative to \( A(3,1) \).

### Problem 6:
1. Given \( S = \{ (x, y) \mid y = \frac{1}{y} \} \), \( A(2,0) \), and \( t \) as the x-axis.
   - **6a.** Find \( k \) if \( C(k,6) \in S' = M_t H_A(S) \).
   - **6b.** Write the equation for \( S' \).

**Solution for 6:**
- For this problem, we need to:
   1. Reflect \( S \) across \( A(2,0) \).
   2. Reflect the result across the x-axis.
   - Solving for \( k \) when \( C(k,6) \in S' \) will involve substituting coordinates and solving.

### Problem 7:
1. Given \( C(2,-1) \), \( Z = \{ (x, y) \mid y = x \} \), and \( t = \{ (x, y) \mid y = 3 - 2x \} \).
   - **7.** Write the equation for \( s = H_C M_z (t) \).

**Solution for 7:**
- For this part, we need to:
   1. Reflect \( t \) across line \( Z \).
   2. Reflect the result across point \( C(2, -1) \).

Would you like a detailed breakdown of each transformation and solution? 

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### Related Questions

1. How do we find the reflection of a curve across a specific point?
2. What transformations are required to reflect a function across a line?
3. How do we determine if a point lies on a reflected curve?
4. What methods are there for reflecting functions across axes and points?
5. How can reflection transformations affect the original shape of a graph?

### Tip
When working with transformations, understanding the basic rules for reflections (across points and lines) is crucial to apply them correctly on various curves and shapes.

5. Jika S = {(x,y) | y = x^2} dan A(3,1)
   a. Apakah B(3,-7) ∈ S' = H_A(S)?
   b. Tulislah persamaan untuk S'
6. Jika S = {(x,y) | y = 1/y} dan A(2,0) dan t sebagai sumbu x.
   a. Carilah nilai k jika C(k,6) ∈ S' = M_t H_A(S)
   b. Tulislah persamaan untuk S'
7. Jika C(2,-1), z = {(x,y) | y = x} dan t = {(x,y) | y = 3 - 2x}, tulislah persamaan untuk s = H_C M_z(t)

Explore solutions for reflection transformations in analytic geometry. Learn how to determine if points belong to reflected sets, write equations for reflected functions, and perform reflections across points and lines. Suitable for high school students studying transformations and function reflections.

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