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.