Question
A ball is thrown off a low cliff to the beach below. It is thrown into the air with an
initial upward velocity of 5 meters per second. The following graph models its height
above sea level in meters over time in seconds.
(x) 2, (y) 10.
A. Would a linear function, a quadratic function, or an exponential function be
most appropriate as a model for this graph? Explain your choice.
initial upward velocity of 5 meters per second. The following graph models its height
above sea level in meters over time in seconds.
(x) 2, (y) 10.
A. Would a linear function, a quadratic function, or an exponential function be
most appropriate as a model for this graph? Explain your choice.
Answers
GPT-4o mini
To determine the most appropriate type of function for modeling the height of the ball above sea level as it is thrown off a low cliff, we should consider the physical behavior of the ball's motion.
1. **Nature of Motion**: The problem describes a ball thrown upward with an initial velocity, which means its motion is affected by gravity. Under the influence of gravity, the height of an object thrown upward follows a parabolic trajectory.
2. **Graph Shape**: The height over time will increase initially (as the ball rises), then peak at a certain height, and subsequently decrease as the ball falls back down. This creates a curve that opens downward, characteristic of a quadratic function.
3. **Quadratic Function**: The general form of a quadratic function is \( h(t) = at^2 + bt + c \), where:
- \( a \) determines the direction of the parabola (downward if negative),
- \( b \) is related to the initial velocity,
- \( c \) is the initial height from which the ball is thrown.
Given that the ball is initially thrown upward and then experiences the force of gravity, a quadratic function is suitable because its path can be accurately represented by a polynomial of degree 2.
4. **Linear and Exponential Functions**:
- **Linear Function**: A linear function represents constant change, which does not occur in this scenario, as the height of the ball changes in a non-linear way due to gravitational acceleration.
- **Exponential Function**: An exponential function would represent processes where growth or decay occurs at a constant relative rate, which also doesn’t apply to the motion of a thrown ball under gravity.
**Conclusion**: A quadratic function is most appropriate for modeling the height of the ball above sea level over time. It accurately reflects the upward motion until the peak height is reached and then the downward motion due to gravity, leading to a parabolic graph shape.
1. **Nature of Motion**: The problem describes a ball thrown upward with an initial velocity, which means its motion is affected by gravity. Under the influence of gravity, the height of an object thrown upward follows a parabolic trajectory.
2. **Graph Shape**: The height over time will increase initially (as the ball rises), then peak at a certain height, and subsequently decrease as the ball falls back down. This creates a curve that opens downward, characteristic of a quadratic function.
3. **Quadratic Function**: The general form of a quadratic function is \( h(t) = at^2 + bt + c \), where:
- \( a \) determines the direction of the parabola (downward if negative),
- \( b \) is related to the initial velocity,
- \( c \) is the initial height from which the ball is thrown.
Given that the ball is initially thrown upward and then experiences the force of gravity, a quadratic function is suitable because its path can be accurately represented by a polynomial of degree 2.
4. **Linear and Exponential Functions**:
- **Linear Function**: A linear function represents constant change, which does not occur in this scenario, as the height of the ball changes in a non-linear way due to gravitational acceleration.
- **Exponential Function**: An exponential function would represent processes where growth or decay occurs at a constant relative rate, which also doesn’t apply to the motion of a thrown ball under gravity.
**Conclusion**: A quadratic function is most appropriate for modeling the height of the ball above sea level over time. It accurately reflects the upward motion until the peak height is reached and then the downward motion due to gravity, leading to a parabolic graph shape.