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.
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.
make it simple in a small paragraph and like a kid wrote it.
Answers
GPT-4o mini
To make a simple equation that shows how the ball goes up and then comes back down, we can use a quadratic function. The general form of a quadratic equation is \( h(t) = at^2 + bt + c \), where \( h(t) \) is the height of the ball, \( t \) is the time in seconds, and \( a \), \( b \), and \( c \) are numbers we need to find. Since the ball was thrown up with an initial speed of 5 meters per second, we can say the ball starts at a height of 0 meters (on the cliff). So we can use \( h(t) = -5t^2 + 5t \). This equation shows that the ball goes up first (because of the \(5t\)) and then comes down (because of the \(-5t^2\)). It makes a nice U-shape like the path of the ball!