Graph 
 
\[h(x)=4\cos\left(\dfrac{\pi}{3}x+\pi\right)-3\] in the interactive widget.
Note that one moveable point always defines an extremum point in the graph and the other point always defines a neighbouring intersection with the midline.
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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.

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

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#### **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?

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#### **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.

Graph h(x) = 4cos((π/3)x + π) - 3 in the interactive widget. Note that one moveable point always defines an extremum point in the graph and the other point always defines a neighboring intersection with the midline.

Explore the graph of the cosine function h(x) = 4cos((π/3)x + π) - 3 with detailed explanations. Understand key properties such as amplitude, period, phase shift, and vertical shift. Learn how to identify extrema and midline intersections, suitable for Grades 10-12.

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