Question
A ball is thrown into the air from the edge of a 48-foot-high cliff so that it eventually lands on the ground. The graph below shows the height, y
𝑦
, of the ball from he ground after x
𝑥
seconds.
What is the range/ Interval notation.
HINT: How to write interval Notation for Quadratic Functions
(1 point)
Responses
0≤x≤2.5
0≤x≤2.5
2.5
2.5<x<5.5
2.5
5.5
2.5<x<5.5
2.5
5.5
0<x<5.5
5.5
0<x<5.5
5.5
x≥2
𝑦
, of the ball from he ground after x
𝑥
seconds.
What is the range/ Interval notation.
HINT: How to write interval Notation for Quadratic Functions
(1 point)
Responses
0≤x≤2.5
0≤x≤2.5
2.5
2.5<x<5.5
2.5
5.5
2.5<x<5.5
2.5
5.5
0<x<5.5
5.5
0<x<5.5
5.5
x≥2
Answers
GPT-4o mini
To determine the range of the height \( y \) of the ball after it is thrown from the 48-foot cliff, we need to clarify a few points based on the typical behavior of a projectile.
1. **Starting Height**: The ball starts at a height of 48 feet (at \( x = 0 \)).
2. **Maximum Height**: The height of the ball will initially increase as it is thrown upwards until it reaches its maximum height.
3. **Landing**: Eventually, the ball will hit the ground, which is at a height of 0 feet.
Assuming that the graph is a parabola that opens downwards (as is common for projectile motion), the ball will have a maximum height above 48 feet and will return to the ground at height \( y = 0 \).
### Interval Notation
Given the nature of the problem and the typical behavior of a projectile:
- The **maximum height** (let's denote it as \( h \)) could possibly be more than 48 feet, depending on the specifics of the initial velocity.
- The lowest point is \( 0 \) feet when the ball hits the ground.
Thus, the **range** can be represented in interval notation.
The range \( y \) of the height function would typically be something like:
\[
[0, h]
\]
Where \( h \) is the maximum height achieved by the ball, which would be greater than 48 feet and could be determined from the graph. Without the specific maximum height, let's assume \( h > 48 \).
### Conclusion
If the graph shows that the ball reaches a maximum height at some point and then descends back to 0, the final range in interval notation would look like:
- If the maximum height is, for example, more than 48, the range would be:
\[
[0, h]
\]
However, if we have the information that \( h \) equals 5.5 when the ball's range is given as higher than the initial 48, your answer would need to reflect that.
Considering your responses, the correct choice would be based on the stated heights from your options. If maximum \( y \) is at some value above zero and indeed at the maximum or specific to the situation.
So your range option could be \( [0, 5.5] \) if we precisely know that the maximum height achieved is \( 5.5 \) or maximum say, just above \( x \) indicates permissible heights as given. Nonetheless, ensure checking the options that correlate with actual graph specifics.
1. **Starting Height**: The ball starts at a height of 48 feet (at \( x = 0 \)).
2. **Maximum Height**: The height of the ball will initially increase as it is thrown upwards until it reaches its maximum height.
3. **Landing**: Eventually, the ball will hit the ground, which is at a height of 0 feet.
Assuming that the graph is a parabola that opens downwards (as is common for projectile motion), the ball will have a maximum height above 48 feet and will return to the ground at height \( y = 0 \).
### Interval Notation
Given the nature of the problem and the typical behavior of a projectile:
- The **maximum height** (let's denote it as \( h \)) could possibly be more than 48 feet, depending on the specifics of the initial velocity.
- The lowest point is \( 0 \) feet when the ball hits the ground.
Thus, the **range** can be represented in interval notation.
The range \( y \) of the height function would typically be something like:
\[
[0, h]
\]
Where \( h \) is the maximum height achieved by the ball, which would be greater than 48 feet and could be determined from the graph. Without the specific maximum height, let's assume \( h > 48 \).
### Conclusion
If the graph shows that the ball reaches a maximum height at some point and then descends back to 0, the final range in interval notation would look like:
- If the maximum height is, for example, more than 48, the range would be:
\[
[0, h]
\]
However, if we have the information that \( h \) equals 5.5 when the ball's range is given as higher than the initial 48, your answer would need to reflect that.
Considering your responses, the correct choice would be based on the stated heights from your options. If maximum \( y \) is at some value above zero and indeed at the maximum or specific to the situation.
So your range option could be \( [0, 5.5] \) if we precisely know that the maximum height achieved is \( 5.5 \) or maximum say, just above \( x \) indicates permissible heights as given. Nonetheless, ensure checking the options that correlate with actual graph specifics.