Let's solve the problems based on the graph and table provided in the images.

1. Where was the ball kicked from?

The ball was kicked from the ground, which corresponds to a height of 0 feet at time $x = 0$ seconds.

• Answer: The ball was kicked from the ground at $x = 0$, $f(x) = 0$.

2. How long is the ball's height increasing?

The height of the ball increases until it reaches its maximum point, which occurs at $x = 2.5$ seconds (from the table).

• Answer: The ball's height is increasing for $2.5$ seconds.

3. What is the maximum height of the ball?

From the table, the maximum height is $f(x) = 37.5$ feet at $x = 2.5$ seconds.

• Answer: The maximum height of the ball is 37.5 feet.

4. When does the ball reach the ground?

The ball reaches the ground when its height is $0$. This happens at:

• $x = 0$ seconds (initial point), and
• $x = 5$ seconds (from the table and graph).

Thus, the ball reaches the ground at $x = 5$ seconds.

• Answer: The ball reaches the ground at $x = 5$ seconds.

Final Summary:

1. Ball was kicked from: the ground (0 feet, $x = 0$).
2. Height increases for: 2.5 seconds.
3. Maximum height: 37.5 feet.
4. Ball reaches the ground at: 5 seconds.

Would you like a step-by-step explanation for any of these? Let me know!

Related Questions:

1. At what time does the ball reach half of its maximum height?
2. Write the equation of the parabola representing the ball's motion.
3. What is the average rate of change of the ball's height from $x = 0$ to $x = 2.5$?
4. What is the symmetry line of the parabola?
5. How would the motion change if the ball reached a higher maximum height?

Tip: The vertex of a parabola represents the maximum or minimum point, depending on its orientation! For a "downward" parabola like this one, the vertex gives the maximum height.