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?

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.