The graph shows how the velocity of a ball changes over time after it is thrown into the air. The graph's velocity $v$ is measured in feet per second on the vertical axis, and time $t$ is in seconds on the horizontal axis.

Analyzing the Graph:

1. Initial Velocity: At time $t = 0$, the velocity $v = 128$ feet per second (this is the y-intercept).
2. Final Velocity: At time $t = 4$, the velocity $v = 0$ feet per second (the ball reaches its peak height and its velocity becomes zero).
3. Negative Slope: The graph has a negative slope, meaning the velocity decreases uniformly as time increases.

Slope Calculation:

• The slope of the line can be calculated as the change in velocity over the change in time:

$\text{slope} = \frac{\Delta v}{\Delta t} = \frac{0 - 128}{4 - 0} = \frac{-128}{4} = -32 \, \text{feet per second per second}.$

This tells us that the velocity is decreasing by 32 feet per second for each second.

Equation of the Line:

Using the point-slope form $v = v_0 + mt$, where:

• $v_0 = 128$ (initial velocity),
• $m = -32$ (slope).

The equation becomes: $v = 128 - 32t.$

Answer:

The correct equation is: $v = 128 - 32t.$

This matches the third option in the provided choices.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How would the graph look if the initial velocity were doubled?
2. What does the slope of the velocity vs. time graph represent physically?
3. How long would it take for the ball to reach a velocity of $-64$ feet per second?
4. What would the graph look like for a ball thrown downward instead of upward?
5. How does air resistance affect the shape of the velocity vs. time graph?

Tip: The area under a velocity-time graph gives the displacement or distance traveled over time.