A basketball thrown from the edge of the court to the goal
shooter is described by the formula h = −t
2 + 6t + 1,
where h is the height of the basketball in metres after
t seconds.
a. What was the height of the ball when it was first thrown?
b. What was the height of the basketball after 2 s?
c. Show that the ball was first at a height of 6 m above the
ground at 1 second.
d. During which time interval was the ball above a height
of 9 m?
e. Plot the path of the basketball.
f. What was the maximum height of the basketball during
its flight?
g. How long was the ball in flight if it was caught at a
height of 1 m above the ground on its downward path?

### Part (d): During which time interval was the ball above a height of 9 meters?

The ball was above 9 meters between the time intervals \( t_1 = 2 \) seconds and \( t_2 = 4 \) seconds.

### Part (e): Plot the path of the basketball

The graph displayed shows the path of the basketball as it travels over time. It also highlights key points:
- The ball crosses 9 meters between 2 and 4 seconds (red line).
- The ball returns to 1 meter at the end of its flight (green line).
- The maximum height is reached at 3 seconds (purple point).

### Part (f): What was the maximum height of the basketball during its flight?

The maximum height of the basketball during its flight was **10 meters**, occurring at \( t = 3 \) seconds.

### Part (g): How long was the ball in flight if it was caught at a height of 1 meter above the ground on its downward path?

The ball was in flight for **6 seconds**, as it returned to a height of 1 meter at \( t = 6 \) seconds.

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Do you have any questions or need more details?

Here are five related questions:
1. How would changing the initial velocity affect the height of the basketball?
2. What is the significance of the vertex of the parabola in projectile motion?
3. How could we find the exact speed of the basketball at any time \( t \)?
4. What would the equation look like if the basketball were thrown from a different height?
5. How could wind resistance or air drag affect the path of the ball?

**Tip:** The vertex form of a quadratic function can quickly give you the maximum or minimum value without needing to graph the function.

A basketball thrown from the edge of the court to the goal shooter is described by the formula h = −t^2 + 6t + 1, where h is the height of the basketball in metres after t seconds. a. What was the height of the ball when it was first thrown? b. What was the height of the basketball after 2 s? c. Show that the ball was first at a height of 6 m above the ground at 1 second. d. During which time interval was the ball above a height of 9 m? e. Plot the path of the basketball. f. What was the maximum height of the basketball during its flight? g. How long was the ball in flight if it was caught at a height of 1 m above the ground on its downward path?

Explore how to analyze the height of a basketball using the quadratic equation h = -t^2 + 6t + 1. This problem demonstrates key concepts in algebra and projectile motion, including finding maximum height, time intervals, and plotting the path of the ball. Ideal for students in Grades 9-12.

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