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The equation provided is \( y = 3|x - 4| + 5 \). This is an absolute value function, which typically has a V-shape graph.

### Key Features to Graph:
1. **Vertex**: The absolute value function \( y = a|x - h| + k \) has its vertex at \( (h, k) \). Here, \( h = 4 \) and \( k = 5 \), so the vertex is at \( (4, 5) \).
2. **Slope**: The coefficient of the absolute value, \( a = 3 \), determines the slope of the two lines forming the V-shape. The slope is positive 3 for both sides of the vertex.
3. **Shape**: The function is symmetric around \( x = 4 \), with a slope of \( 3 \) for the right side (increasing) and \( -3 \) for the left side (decreasing).

### Steps to Plot:
1. **Plot the vertex** at \( (4, 5) \).
2. **For the right side**: 
   - For \( x = 5 \), \( y = 3|5 - 4| + 5 = 8 \) (point \( (5, 8) \)).
   - For \( x = 6 \), \( y = 3|6 - 4| + 5 = 11 \) (point \( (6, 11) \)).
3. **For the left side**:
   - For \( x = 3 \), \( y = 3|3 - 4| + 5 = 8 \) (point \( (3, 8) \)).
   - For \( x = 2 \), \( y = 3|2 - 4| + 5 = 11 \) (point \( (2, 11) \)).
4. Draw lines connecting these points. The right side of the vertex increases with a slope of 3, and the left side decreases with a slope of -3.

### Graph Summary:
- The vertex at \( (4, 5) \) is the lowest point (since the coefficient \( a = 3 \) is positive).
- The graph opens upwards in a V-shape, with both branches having slopes of \( \pm 3 \).

Would you like further details on any step, or assistance with more problems related to absolute value graphs?

### Related Questions:
1. How does changing the coefficient of \( x \) affect the steepness of the graph?
2. What happens to the graph if \( h \) or \( k \) are altered?
3. How would the graph look if the absolute value term was negative (e.g., \( y = -3|x - 4| + 5 \))?
4. Can you explain why absolute value graphs are always V-shaped?
5. How do transformations such as shifting affect other types of functions, like quadratic or linear?

### Tip:
When graphing an absolute value function, always start by locating the vertex and then calculate a few points on either side to understand the shape.

This problem covers graphing the absolute value function y = 3|x - 4| + 5. Learn how to identify the vertex and slope, and how to graph this V-shaped function. Perfect for students in Grades 8-10.

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