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. Would a linear function, a quadratic function, or an exponential function be most appropriate as a model for this graph? Explain your choice. already done.
The best way to model the height of the ball is with a quadratic function. This is because when you throw something up, it goes up to a peak and then comes back down, which makes a U-shaped graph. A linear function would look like a straight line, and an exponential function goes up really fast, like a rocket, which isn't what happens here. So, a quadratic function shows how the ball goes up, stops, and then falls back down, just like the graph.
Use your table to create an equation that models this graph and scenario. already done.
Interpret the meaning of the y-intercept of this graph in terms of the given context. already done.
The y-intercept of the graph is the point where the line crosses the y-axis, which happens when time is zero. In this situation, it tells us how high the ball is when it is first thrown off the cliff. So, the y-intercept shows the starting height of the ball above sea level when the person throws it. If the y-intercept is, for example, 10 meters, that means the ball is 11 meters above the ground when it's thrown.
Write your own question that asks a student to interpret a different key feature of this graph in terms of the given context. Then, provide the solution for your question.
make it simple in a small paragraph and like a kid wrote it.
The best way to model the height of the ball is with a quadratic function. This is because when you throw something up, it goes up to a peak and then comes back down, which makes a U-shaped graph. A linear function would look like a straight line, and an exponential function goes up really fast, like a rocket, which isn't what happens here. So, a quadratic function shows how the ball goes up, stops, and then falls back down, just like the graph.
Use your table to create an equation that models this graph and scenario. already done.
Interpret the meaning of the y-intercept of this graph in terms of the given context. already done.
The y-intercept of the graph is the point where the line crosses the y-axis, which happens when time is zero. In this situation, it tells us how high the ball is when it is first thrown off the cliff. So, the y-intercept shows the starting height of the ball above sea level when the person throws it. If the y-intercept is, for example, 10 meters, that means the ball is 11 meters above the ground when it's thrown.
Write your own question that asks a student to interpret a different key feature of this graph in terms of the given context. Then, provide the solution for your question.
make it simple in a small paragraph and like a kid wrote it.
Answers
GPT-4o mini
**Question:** What does the vertex of the graph represent in the context of the ball being thrown off the cliff?
**Solution:** The vertex of the graph is the highest point that the ball reaches after being thrown. This means it tells us how high the ball goes before it starts to come back down. So, if the vertex is at 15 meters, it means the ball goes up to 15 meters in the air before it starts falling down to the beach. It’s like when you jump up high and then come down; the vertex is your highest jump!
**Solution:** The vertex of the graph is the highest point that the ball reaches after being thrown. This means it tells us how high the ball goes before it starts to come back down. So, if the vertex is at 15 meters, it means the ball goes up to 15 meters in the air before it starts falling down to the beach. It’s like when you jump up high and then come down; the vertex is your highest jump!