It seems you're requesting a graph of the function $h(x) = 4\cos\left(\frac{\pi}{3}x + \pi\right) - 3$, and you also want interactive features where one point identifies an extremum and another marks a neighboring intersection with the midline.

Since I can't directly create an interactive widget here, I'll describe how you can graph it and simulate this interaction step-by-step, or use software tools like Desmos or GeoGebra for a fully interactive experience.

---

Steps to Graph $h(x) = 4\cos\left(\frac{\pi}{3}x + \pi\right) - 3$:

1. Key Properties of the Function:

• Amplitude: $4$ (distance from the midline to a peak or trough).
• Midline: $y = -3$ (the function oscillates around this line).
• Period: The period is given by $\frac{2\pi}{\frac{\pi}{3}} = 6$.
• Phase Shift: The phase shift is $-\frac{\pi}{\frac{\pi}{3}} = -3$ (shift to the left by 3 units).
• Vertical Shift: The graph is shifted down by 3 units.

2. Midline and Extrema:

• Midline Intersection: Points where $h(x) = -3$. Solve: $4\cos\left(\frac{\pi}{3}x + \pi\right) - 3 = -3$ This simplifies to $\cos\left(\frac{\pi}{3}x + \pi\right) = 0$.
• Extrema: Maximum $y = 1$ and Minimum $y = -7$.

3. Interactive Points:

• Place a point at the extrema $x$ values. These occur when $\cos\left(\frac{\pi}{3}x + \pi\right) = \pm1$.
• A second point defines intersections with the midline ($y = -3$).

4. Suggested Tool:

• Use Desmos or GeoGebra:
  • Input $h(x) = 4\cos\left(\frac{\pi}{3}x + \pi\right) - 3$.
  • Add sliders for interactive points marking extrema and midline intersections.

---

Would you like a static graph plotted or details on using software for interactivity?

---

Follow-Up Questions:

1. Would you like me to explain how the phase shift affects the graph visually?
2. Should I calculate the exact extrema points and midline intersections for you?
3. Would you like additional examples of cosine graphs with varying parameters?
4. Do you need a detailed explanation of the cosine function properties (e.g., amplitude and period)?
5. Would you like help setting up this graph in a specific tool like Desmos?

---

Tip:

To easily visualize periodic functions, always mark the midline, extrema, and key points (e.g., intersections) first, then sketch the shape using the period and amplitude.