Problem Breakdown:

We are tasked with finding the coordinates of the point $P(3, -2)$ after applying three different transformations to the function $y = f(x)$.

Part (i): $y = f(x - 2)$

• Transformation type: Horizontal translation.
• Effect: Shifts the graph 2 units to the right.
• New $x$-coordinate: Add 2 to $x$.
  $x_{\text{new}} = 3 + 2 = 5$
• $y$-coordinate remains unchanged: $y_{\text{new}} = -2$.
• Transformed coordinates: $(5, -2)$.

Part (ii): $y = f(2x)$

• Transformation type: Horizontal compression by a factor of $\frac{1}{2}$.
• Effect: Divide $x$-coordinate by 2.
  $x_{\text{new}} = \frac{3}{2} = 1.5$
• $y$-coordinate remains unchanged: $y_{\text{new}} = -2$.
• Transformed coordinates: $(1.5, -2)$.

Part (iii): $y = 3f(-x) + 5$

• Step 1: Reflection in the $y$-axis.
  • Change the sign of $x$:
    $x_{\text{new}} = -3$
• Step 2: Vertical scaling by 3.
  • Multiply the $y$-coordinate by 3:
    $y_{\text{scaled}} = 3 \times (-2) = -6$
• Step 3: Vertical translation upward by 5.
  • Add 5 to the scaled $y$-coordinate:
    $y_{\text{new}} = -6 + 5 = -1$
• Transformed coordinates: $(-3, -1)$.

Final Answers:

1. $(5, -2)$
2. $(1.5, -2)$
3. $(-3, -1)$

Let me know if you'd like a detailed explanation of transformations or have further questions! 😊

Follow-up Questions:

1. How do horizontal translations affect $x$-coordinates in general?
2. Why does a horizontal compression involve dividing the $x$-coordinate?
3. What is the significance of reflections in $x$- or $y$-axes for transformations?
4. How does vertical scaling influence the $y$-coordinate?
5. Can you combine multiple transformations into one equation?

Tip: Always apply transformations in the correct order and carefully consider the sign changes for reflections.