The table shows time and height data representing the motion of an osprey during its dive. The task is to plot these values and determine the best model for the function.

Let's review the data first:

Plotting the Data

To plot these points:

• The x-axis will represent the time in seconds.
• The y-axis will represent the height in meters.

After plotting the data, you would observe that the heights increase initially, reach a peak, and then decrease as time progresses. This suggests a parabolic trend.

Best Model:

The behavior of the data is indicative of a quadratic function. Quadratic functions, which have the general form $y = ax^2 + bx + c$, often model situations where there is a rise to a maximum (or minimum) point followed by a decrease. The bird starts at a certain height, reaches a peak, and then descends, which is typical of parabolic motion.

Next steps:

• Do you have graph paper or graphing software to plot the points?
• Would you like me to fit a quadratic equation to these data points or help further with the graph?

Questions:

1. What characteristics make quadratic functions suitable for modeling such motion?
2. How do you think air resistance might affect the height over time?
3. Can a polynomial of degree higher than 2 fit this data better?
4. Would adding more data points (e.g., at 1-second intervals) provide a better fit for the function?
5. How can you determine the vertex of the quadratic function from the given data?

Tip: Quadratic models often describe real-world phenomena involving acceleration and deceleration, such as projectile motion.