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Solve using higher maths and sketch graphs 

The problem involves understanding and sketching graphs based on the given conditions.

### Problem Summary:
1. **Given Function:** \( y = g(x) \) (part of the graph is shown).
2. **Key Points:** The function has stationary points at \( x = 1 \) and \( x = 3 \), and this is reflected in the graph provided.
3. **Tasks:** 
   - Sketch the graph of the related functions:
     - \( y = g(-x) + 2 \)
     - \( y = -g(x - 1) \)

### Solution Steps:

#### 1. **Understanding Transformations:**
   - **\( y = g(-x) + 2 \):**
     - **Reflection:** The function is reflected across the y-axis due to the \( g(-x) \).
     - **Vertical Shift:** The entire graph is shifted 2 units up due to the \( +2 \).

   - **\( y = -g(x - 1) \):**
     - **Horizontal Shift:** The function is shifted 1 unit to the right due to the \( (x - 1) \).
     - **Reflection:** The function is reflected across the x-axis due to the \( -g(x) \).

#### 2. **Sketching the Graphs:**
   - **\( y = g(-x) + 2 \):**
     - Start by reflecting the original graph across the y-axis.
     - Then, shift the entire graph 2 units upwards.
  
   - **\( y = -g(x - 1) \):**
     - Start by shifting the original graph 1 unit to the right.
     - Then, reflect the graph across the x-axis.

### Graph Sketches:

- **For \( y = g(-x) + 2 \):**
  - Reflect the graph horizontally and then move it up.
  - The stationary points that were originally at \( x = 1 \) and \( x = 3 \) would now be at \( x = -1 \) and \( x = -3 \) respectively, and their \( y \)-coordinates would be increased by 2.

- **For \( y = -g(x - 1) \):**
  - Shift the graph 1 unit to the right, then reflect it vertically.
  - The stationary points that were originally at \( x = 1 \) and \( x = 3 \) would be moved to \( x = 2 \) and \( x = 4 \) respectively, with their \( y \)-coordinates being inverted.

These transformations can be applied step by step to ensure that the graphs are accurately sketched. 

Would you like the specific graph sketch details for these transformations?

Explore how to understand and sketch graphs using transformations. This problem involves sketching graphs based on given conditions and applying transformations such as reflection, vertical and horizontal shifts. Suitable for advanced high school students studying graph transformations.

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