The problem describes a ball thrown from a cliff that is 48 feet above the ground. The ball reaches a maximum height of 64 feet and hits the ground 3 seconds after being thrown. We are tasked with identifying which graph corresponds to the quadratic function $f(t)$ modeling this motion.

Step-by-Step Breakdown:

1. Initial Height (at $t = 0$):
   The ball is thrown from a cliff 48 feet above the ground. So, $f(0) = 48$. The graph must start at a height of 48 feet when $t = 0$.

2. Maximum Height:
   The ball reaches a maximum height of 64 feet. This should occur before hitting the ground (at some time $t$) but we know the total time is 3 seconds.

3. Time to Hit the Ground:
   The ball hits the ground after 3 seconds, so $f(3) = 0$. The graph must end at a height of 0 feet at $t = 3$.

Graph Identification:

• We need a parabola that starts at 48 feet, reaches a maximum height of 64 feet, and falls to 0 feet at $t = 3$.

Based on the graphs shown in your image:

• The top left graph doesn't satisfy the condition as the ball doesn't reach the maximum height.
• The top right graph reaches the ground before 3 seconds, so it's incorrect.
• The bottom left graph seems to satisfy all conditions: it starts at 48 feet, peaks at 64 feet, and hits the ground at 3 seconds.
• The bottom right graph shows the wrong motion as the ball doesn't reach the maximum height of 64 feet.

Conclusion:

The bottom left graph represents the correct function.

Would you like further details or clarification on any of the steps? Here are some related questions to consider:

1. What is the general form of the quadratic function for projectile motion?
2. How can you find the vertex of a quadratic function?
3. How does the initial velocity affect the height of the ball?
4. What role does gravity play in determining the shape of the parabola?
5. How would the graph change if the cliff height was different?

Tip: The vertex form of a quadratic equation is very useful when dealing with maximum or minimum values in real-life applications like projectile motion.