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To solve this problem, we need to graph and write equations for a continuous path for Tony Hawk from point T (-4, 1) to point P (14, 16) while avoiding the garbage cans located at points A, B, C, D, E, F, and G. The path must include at least two parabolas (one opening up and one down), and at least two lines.

### Step-by-Step Plan

1. **Identify the coordinates of the garbage cans:**
   - A: (-3, 2)
   - B: (8, 6)
   - C: (0, 5)
   - D: (4, 12)
   - E: (14, 13)
   - F: (11, 15)
   - G: (7, 3)

2. **Define the path:**
   - We need to use a combination of parabolas and lines.
   - We can split the path into sections where each section is either a parabola or a line.

3. **Write the equations:**
   - Ensure the equations avoid the given points.
   - Include the required features (parabolas with different orientations and at least one stretch factor).

### Sketching the Path

We'll start by defining a path in four segments:

1. **Parabola Opening Up (Segment 1):**
   - Start at T (-4, 1) and make sure it opens up and avoids garbage cans.
   - Vertex should be somewhere along the y-axis for simplicity, say (-1, -1).
   - General form: \( y = a(x - h)^2 + k \).

2. **Line Segment (Segment 2):**
   - Connect the end of the first parabola to an intermediate point, ensuring it avoids garbage cans.
   - Use linear equation form \( y = mx + b \).

3. **Parabola Opening Down (Segment 3):**
   - Connect to the next point and ensure it opens down.
   - Similar to the first parabola, but opening down.
   - General form: \( y = -a(x - h)^2 + k \).

4. **Line Segment (Segment 4):**
   - Final segment connecting to P (14, 16).

### Path Equations

Let's determine the exact equations for each segment.

**Segment 1: Parabola Opening Up**
- Choose vertex (0, -1).
- Equation form: \( y = a(x - 0)^2 - 1 \).
- We need two points on this parabola. Let’s choose (-4, 1) and another point that avoids the garbage cans.

**Segment 2: Line Segment**
- Connect the end of the first parabola to the next point. Let’s assume it ends at (4, 7).
- Equation of line: Find slope and intercept.

**Segment 3: Parabola Opening Down**
- Choose vertex (6, 12).
- Equation form: \( y = -a(x - 6)^2 + 12 \).
- Choose points on this parabola avoiding garbage cans.

**Segment 4: Line Segment**
- Connect to P (14, 16).
- Equation of line: Find slope and intercept.

### Detailed Equations

1. **Segment 1:**
   \[ y = \frac{1 - (-1)}{(-4 - 0)^2} (x - 0)^2 - 1 \]
   \[ y = \frac{2}{16}x^2 - 1 \]
   \[ y = \frac{1}{8}x^2 - 1 \]

2. **Segment 2:**
   - Connecting (4, 7) to (8, 6).
   - Slope \( m = \frac{6 - 7}{8 - 4} = -\frac{1}{4} \).
   - Equation: \( y = -\frac{1}{4}x + b \).
   - Using point (4, 7):
   \[ 7 = -\frac{1}{4}(4) + b \]
   \[ 7 = -1 + b \]
   \[ b = 8 \]
   \[ y = -\frac{1}{4}x + 8 \]

3. **Segment 3:**
   \[ y = -a(x - 6)^2 + 12 \]
   - Choose two points, e.g., (10, 6).

4. **Segment 4:**
   - Connect (10, 6) to (14, 16).
   - Slope \( m = \frac{16 - 6}{14 - 10} = \frac{10}{4} = 2.5 \).
   - Equation: \( y = 2.5x + b \).
   - Using point (10, 6):
   \[ 6 = 2.5(10) + b \]
   \[ 6 = 25 + b \]
   \[ b = -19 \]
   \[ y = 2.5x - 19 \]

Would you like more details or have any questions about these equations? 

Here are 5 questions you might ask:
1. How do you find the vertex form of a parabola?
2. How can we ensure the path avoids all garbage cans?
3. What is the significance of the stretch factor in a parabola?
4. How do you convert between different forms of quadratic equations?
5. What are the properties of a line equation?

**Tip:** Always verify your chosen points on the path to ensure they avoid the specified obstacles.

Explore how to graph a continuous path for Tony Hawk using lines and parabolas, avoiding obstacles. This problem covers concepts like geometry, equations of lines, and quadratic equations. Suitable for high school students.

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