The graph shown represents the equation:

$g(x) = 5 |x - 6| + 2$

Let’s break this down step by step:

Understanding the Equation

1. General Form: $g(x) = a |x - h| + k$, where:

   • $a$ controls the slope (steepness).
   • $(h, k)$ is the vertex (the turning point of the graph).
   • The absolute value causes the "V" shape.

2. Key Information:

   • $a = 5$: This gives the slope of the two "arms" of the graph.
   • $h = 6$ and $k = 2$: The vertex of the graph is at $(6, 2)$.

Graph Characteristics

1. Vertex:

   • The graph changes direction at the vertex $(6, 2)$.

2. Slopes:

   • To the right of $x = 6$, the slope is $+5$.
   • To the left of $x = 6$, the slope is $-5$.

3. Shape:

   • The "V" shape is symmetric about the vertical line $x = 6$.

Analyzing the Graph

• The graph begins at $(6, 2)$ and extends upward symmetrically.
• The positive slope causes the right arm to increase steeply.
• The negative slope creates the downward trend on the left arm.

Key Observations from the Graph

• The green dot at $(6, 2)$ marks the vertex.
• Both arms extend linearly with a steepness corresponding to $5$.

Would you like a detailed explanation of how this equation was graphed step by step? Let me know!

Related Questions to Explore

1. How does changing the value of $a$ affect the graph of an absolute value function?
2. What happens if the vertex of $g(x) = 5|x - h| + k$ moves to $(4, 3)$?
3. How do you determine the domain and range of absolute value functions?
4. How would the graph change if the equation were $g(x) = -5 |x - 6| + 2$?
5. Can you calculate $g(8)$ and $g(4)$ for this function?

Tip: To graph absolute value functions, always identify the vertex and the slopes of the arms. This ensures accuracy when sketching the "V" shape.