It looks like you've uploaded an image containing a graph of the position of a particle over time, along with some related questions. Let's break it down and solve them step by step.

1. Understanding the Graph:

   • The graph represents the position $x(t)$ of a particle as a function of time $t$.
   • The horizontal axis is time $t$ (in seconds), and the vertical axis is position $x$ (in meters).

2. First Question:

   • "What is the particle's average velocity between times $t = 0 \, \text{s}$ and $t = 8 \, \text{s}?$"

The average velocity $v_{\text{avg}}$ is defined as:

$v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x(t_{\text{final}}) - x(t_{\text{initial}})}{t_{\text{final}} - t_{\text{initial}}}$

Let's calculate the change in position between $t = 0 \, \text{s}$ and $t = 8 \, \text{s}$ based on the graph and then compute the average velocity.

3. Second Question:
   • "What is the average speed of the particle between times $t = 6 \, \text{s}$ and $t = 7 \, \text{s}?$"

The average speed is the total distance traveled (regardless of direction) divided by the total time.

4. Third Question:
   • "What is the average acceleration of the particle in the time interval between $t = 1.5 \, \text{s}$ and $t = 4 \, \text{s}?$"

Average acceleration $a_{\text{avg}}$ is defined as:

$a_{\text{avg}} = \frac{\Delta v}{\Delta t}$

We will analyze the velocity changes between $t = 1.5 \, \text{s}$ and $t = 4 \, \text{s}$.

I'll extract the values from the graph and perform the calculations next.