IM Alg1.6.6 Lesson: Building Quadratic Functions to Describe Situations (Part 2)

A cannon is 10 feet off the ground. It launches a cannonball straight up with a velocity of 406 feet per second.

[size=150]Imagine that there is no gravity and that the cannonball continues to travel upward with the same velocity.[/size]

Complete the table with the heights of the cannonball at different times.

Write an equation to model the distance in feet, [math]d[/math], of the ball [math]t[/math] seconds after it was fired from the cannon if there was no gravity.[br]

Earlier, you completed a table that represents the height of a cannonball, in feet, as a function of time, in seconds, if there was no gravity.

[size=150]This table shows the actual heights of the ball at different times.[/size][br][center][img]data:image/png;base64,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[/img][/center][br]Compare the values in this table with those in the table you completed earlier. Make at least 2 observations.

Plot the two sets of data you have on the same coordinate plane. Right click on a point to find the setting to change its color.

How are the two graphs alike? How are they different?[br]

Write an equation to model the actual distance [math]d[/math], in feet, of the ball [math]t[/math] seconds after it was fired from the cannon. If you get stuck, consider the differences in distances and the effects of gravity from a previous lesson.[br]

[size=150]The function defined by [math]d=50+312t-16t^2[/math] gives the height in feet of a cannonball [math]t[/math] seconds after the ball leaves the cannon.[/size][br][br][size=100]What do the terms 50, [math]312t[/math], and [math]-16t^2[/math] tell us about the cannonball?[/size][br]

Use graphing technology to graph the function. Adjust the graphing window to the following boundaries: 0 < x < 25 and 0 < y < 2,000.

Observe the graph and:

Describe the shape of the graph. What does it tell us about the movement of the cannonball?[br]

Estimate the maximum height the ball reaches. When does this happen?[br]

Estimate when the ball hits the ground.[br]

What domain is appropriate for this function? Explain your reasoning.[br]

If the cannonball were fired at 800 feet per second, would it reach a mile in height? Explain your reasoning.