Building Quadratic Functions to Describe Situations (Part 2)
Integrated Math 2
6.1
Activity
A person with a t-shirt launcher is standing in the center of the field at a soccer stadium. He is holding the launcher so that the mouth of the launcher, where the t-shirts exit the launcher, is 5 feet above the ground. The launcher sends a t-shirt straight up with a velocity of 90 feet per second.
Imagine that there is no gravity and that the t-shirt continues to travel upward with the same velocity.
Complete the table with the heights of the t-shirt at different times.
seconds
0
1
2
3
4
5
distance above ground (feet)
5
Write an equation to model the distance in feet, , of the t-shirt seconds after it was launched if there was no gravity.
6.2
Activity
Earlier, you completed a table that represents the height of a t-shirt, in feet, as a function of time, in seconds, if there were no gravity.
This table shows the actual heights of the t-shirt at different times.
seconds
0
1
2
3
4
5
distance above ground (feet)
5
79
121
131
109
55
Compare the values in this table with those in the table that you completed earlier. Make at least 2 observations.
Plot the two sets of data that you have on the same coordinate plane.
How are the two graphs alike? How are they different?
Write an equation to model the actual distance, , in feet, of the t-shirt seconds after it was launched. If you get stuck, consider the differences in distances and the effects of gravity from an earlier lesson.
6.3
Activity
The function defined by gives the height in feet of a cannonball seconds after the ball leaves the cannon.
What do the terms 50, , and tell us about the cannonball?
Use graphing technology to graph the function. Adjust the graphing window to show: and .
Observe the graph and:
Describe the shape of the graph. What does it tell us about the movement of the cannonball?
Estimate the maximum height that the ball reaches. When does this happen?
Estimate when the ball hits the ground.
What domain is appropriate for this function, based on the situation? Explain your reasoning.
Student Lesson Summary
In this lesson, we looked at the height of objects that are launched upward and then come back down because of gravity.
An object is thrown upward from a height of 5 feet with a velocity of 60 feet per second. Its height, , in feet, after seconds is modeled by the function .
The linear expression represents the height that the object would have at time if there were no gravity. The object would keep going up at the same speed at which it was thrown. The graph would be a line with a slope of 60, which relates to the constant speed of 60 feet per second.
The expression represents the effect of gravity, which eventually causes the object to slow down, stop, and start falling back again.
Notice the graph intersects the vertical axis at 5, which means that the object was thrown into the air from 5 feet off the ground. The graph indicates that the object reaches its peak height of about 60 feet after a little less than 2 seconds. That peak is the point on the graph where the function reaches a maximum value. At that point, the curve changes direction, and the output of the function changes from increasing to decreasing. We call that point the vertex of the graph.
Here is the graph of .
Graph of the quadratic function
on a coordinate plane, origin
. Horizontal axis scale 0 to 4 by 1’s, labeled “time (seconds)”. Vertical axis scale 0 to 80 by 20’s, labeled “distance above ground (feet)”. Some of the points of this function are (0 comma 5), (1 comma 49), to a maximum near (1 point 9 comma 61 point 2 5) then decreasing through (2 comma 61), (3 comma 41) and (3.8 comma 0).
The graph representing any quadratic function is a special kind of “U” shape called a parabola. You will learn more about the geometry of parabolas in a future lesson. Every parabola has a vertex, because there is a point at which it changes direction—from increasing to decreasing, or the other way around.
The object hits the ground a little before 4 seconds. That time corresponds to the horizontal intercept of the graph. An input value that produces an output of 0 is called a zero of the function. A zero of function is approximately 3.8, because .
In this situation, input values less than 0 seconds or more than about 3.8 seconds would not be meaningful, so an appropriate domain for this function would include all values of between 0 and about 3.8.
The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing, or vice versa. It is the highest or lowest point on the graph.
A zero of a function is an input that results in an output of 0. In other words, if , then is a zero of .
