We are given that the function $h(x)$ is graphed, and the point $(3, -2)$ is on the graph of $y = h(x)$. Now, we need to find which of the points is on the graph of $y = 2h(x) - 1$.

Step-by-step process:

1. Original function information: We know that $h(3) = -2$ because the point $(3, -2)$ is on the graph of $y = h(x)$.

2. Transformation of the function: The new function is $y = 2h(x) - 1$. To find the corresponding point for $x = 3$ in the new function:

   • Start with $h(3) = -2$.
   • Substitute $h(3)$ into the new function: $y = 2h(3) - 1 = 2(-2) - 1 = -4 - 1 = -5.$
   • Thus, the point $(3, -5)$ is on the graph of the transformed function $y = 2h(x) - 1$.

Conclusion:

The correct point that lies on the graph of $y = 2h(x) - 1$ is $(3, -5)$, which corresponds to option B.

Would you like further details on any part of the solution?

Here are 5 related questions to deepen your understanding:

1. How does multiplying a function by a constant affect its graph?
2. What happens when you subtract a constant from a function?
3. How can transformations of functions be identified based on the equation?
4. How would the graph of $y = -2h(x) + 3$ be transformed compared to $y = h(x)$?
5. What is the general form of function transformations, and how can you apply them?

Tip: Always carefully apply transformations step by step to ensure you're correctly interpreting how each change affects the graph!