https://file.aiquickdraw.com/mathbot/file/1730162522952crz0x5vs.jpg

Let's analyze the information from the image and solve the questions based on the graph of the function.

### Problem Breakdown

1. **Identify the Type of Function**: We need to determine if the graph represents a trigonometric (sinusoidal) function, which describes oscillating motion, common for springs.

2. **Initial Position of the Weight**: The initial position is given by the value of \( p(t) \) at \( t = 0 \) seconds, where \( p(0) \) represents the starting position of the weight.

3. **First Time the Weight Reaches Equilibrium**: The equilibrium point occurs where \( p(t) = 0 \), so we need to find the smallest \( t \) value where this is true.

### Observations from the Graph

- The graph shows oscillations typical of a **trigonometric (sinusoidal) function**, likely a cosine function based on the shape.
- **Initial Position** \( p(0) \): At \( t = 0 \), \( p(t) \) is approximately 40 cm.
- **Equilibrium** (where \( p(t) = 0 \)): This first occurs around \( t = 1 \) second.

### Answers

1. **The graph is a sinusoidal function.**
2. **The initial position of the weight is 40 centimeters.**
3. **The weight first reaches equilibrium when \( t = 1 \) second.**

Would you like further details or have any questions?

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Here are some additional questions that extend the concepts involved:

1. What is the frequency of oscillation in this graph?
2. How would the graph change if the spring constant increased?
3. What is the amplitude of the oscillation in this scenario?
4. How can we determine the period of oscillation from this graph?
5. How would damping (energy loss) affect the motion of the spring?

**Tip:** For sinusoidal graphs, the amplitude gives the maximum displacement from equilibrium, and the period represents the time to complete one full oscillation.

A weight is attached to a spring, which moves up and down as a function of time. p(t) gives the position of the weight at time t in centimeters, where t is in seconds. Complete the sentences based on the graph of the function.

Explore the sinusoidal function of a weight attached to a spring, determining initial position, equilibrium points, and oscillation behavior. This problem involves concepts like sinusoidal functions, equilibrium, and oscillatory motion, suitable for high school students in Grades 10-12.

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