1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.4
TASK 3.2.4: PATTERNS OF PROJECTILES — WHAT GOES UP…
Solutions
The graph shown is the height of a ball
as it is thrown from ground level
straight up into the air and then returns
back to the ground four different
times. The general form of the
equation that models these situations is
the quadratic h = ! 12 gt 2 + v0t + h0
where h is the height, t is the time in
seconds, and g is the acceleration of
ft
gravity (on Earth, 32 sec2 ). v0 is
determined by the initial velocity and
h0 by the initial starting height of the
ball. Since we are modeling a
situation on the Earth’s surface, our
formula becomes h = !16t 2 + v0t + h 0 .
1. Note that each of the four parabolas start at the point (0, 0). What does this point
mean in this situation? The point (0, 0) represents that at the beginning of each
toss, at the moment we started tracking time (t = 0 seconds), the height of each
ball was 0 feet.
2.
a. For each graph, determine the length of time that the ball was moving in
an upward direction. Explain your answer.
We can tell how long each ball was traveling in an upward direction by
looking at the location of the vertex. Ball 1 was traveling upward 1
second; ball 2, 1.5 seconds; ball 3, 2 seconds; and ball 4, 2.5 seconds.
b. For each graph, determine the length of time that the ball was moving in a
downward direction. Explain you answer.
We can tell how long that each ball traveled downward by looking at the
x-coordinate of the vertex and the x-intercept of the graph representing the
time that passed before the ball was at a height of 0. Ball 1 was traveling
downward for 1 second; ball 2, 1.5 seconds; ball 3, 2 seconds; and ball 4,
2.5 seconds.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.4
c. What do you notice about the length of time that each ball was moving
upward and the length of time each ball was moving downward?
For each ball, the time it traveled upward is equal to the time it travels
downward.
d. What geometric property does this describe? For each graph, give the
formula for the line that is generated from this property. Explain how
each line is related to the vertex of its corresponding parabola.
This describes the symmetry of the parabola. In particular, there is a line
of symmetry for each parabola. For ball 1, the line of symmetry is y=1;
ball 2, y=1.5; for ball 3, y=2; and for ball 4, y=2.5. For each ball, the
vertex is located at the point (h, k) and the equation for the line of
symmetry is y=k.
e. Will the behavior that you observed in (c) always occur for this situation?
If so, why? If not, what would have to change that would create a
different result?
It is possible that the time that a ball travels upwards is not the same as the
time it travels downward. Participants might come up with a situation
where the ball falls into a hole and thus travels downward more than
upwards. The ball may begin at a different height than 0. Other scenarios
should be examined for accuracy. But if the starting height and the ending
height are the same, the time traveled upward is equal to the time traveled
downward.
3. Examine closely the relationships between the vertices of the graphs. If a 5th and
6th ball are tossed in the same way and their heights follow the patterns
demonstrated by the first four, predict the location of vertices of the 5th and 6th
graphs. Provide support for your prediction. How long will the 5th and 6th balls
remain in the air? Why?
Examining the locations of the vertices and continuing the pattern, it is clear that
the x-coordinates of the vertices of the 5th and 6th balls will be located at 3 and 3.5
respectively. Using finite differences on the heights shows that the maximum
height of the 5th and 6th balls is 144 and 196, respectively. Since the vertex occurs
midway, the 5th and 6th balls will be in the air 6 seconds and 7 seconds,
respectively.
16
36
64
100
20
28
36
8
8
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.4
Looking at the graphs of the height of the four balls, estimate the height of each ball at 2
seconds. The function rule for each of the graphs are given below. Use these rules to
check your estimation by determining the actual height at two seconds.
• 1: h1 = !16t 2 + 32t
Ball 1:
Ball 2:
Ball 3:
Ball 4:
•
2: h2 = !16t 2 + 48t
•
3: h3 = !16t 2 + 64t
•
4: h4 = !16t 2 + 80t
0 feet
32 feet
64 feet
96 feet
5. Using the function rules for the four graphs, predict the rules for the 5th and 6th
graphs that were described in (3). To check your predictions, verify the values
that you found for the vertices of these graphs using your new function rules.
Ball 5: h5 = !16t 2 + 96t ; h5 (3) = !16(3)2 + 96(3) = !144 + 288 = 144
Ball 6: h6 = !16t 2 + 112t ; h6 (3.5) = !16(3.5)2 + 96(3.5) = !196 + 336 = 196
Math notes
One of the confusing aspects of the tossing a ball situation is that it is easy to mistake the
graph of the function as the path of the ball instead of as the relationship between height
and time. Participants may need several experiences to help them really understand the
difference between the graph of the function and the picture of the situation.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.4
TASK 3.2.4: PATTERNS OF PROJECTILES – WHAT GOES UP…
The graph shown is the height of a ball
as it is thrown from ground level
straight up into the air and then returns
back to the ground four different
times. The general form of the
equation that models these situations is
the quadratic h = ! 12 gt 2 + v0t + h0
where h is the height, t is the time in
seconds, and g is the acceleration of
ft
gravity (on Earth, 32 sec2 ). v0 is
determined by the initial velocity and
h0 by the initial starting height of the
ball. Since we are modeling a
situation on the Earth’s surface, our
formula becomes h = !16t 2 + v0t + h 0 .
1. Note that each of the four parabolas start at the point (0, 0). What does this point
mean in this situation?
2.
a. For each graph, determine the length of time that the ball was moving in
an upward direction. Explain your answer.
b. For each graph, determine the length of time that the ball was moving in a
downward direction. Explain you answer.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.4
c. What do you notice about the length of time that each ball was moving
upward and the length of time each ball was moving downward?
d. What geometric property does this describe? For each graph, give the
formula for the line that is generated from this property. Explain how
each line is related to the vertex of its corresponding parabola.
e. Will the behavior that you observed in (c) always occur for this situation?
If so, why? If not, what would have to change that would create a
different result?
3. Examine closely the relationships between the vertices of the graphs. If a 5th and
6th ball are tossed in the same way and their heights follow the patterns
demonstrated by the first four, predict the location of vertices of the 5th and 6th
graphs. Provide support for your prediction. How long will the 5th and 6th balls
remain in the air? Why?
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.4
4. Looking at the graphs of the height of the four balls, estimate the height of each
ball at 2 seconds. The function rules for each of the graphs are given below. Use
these rules to check your estimation by determining the actual height at two
seconds.
• 1: h1 = !16t 2 + 32t
•
2: h2 = !16t 2 + 48t
•
3: h3 = !16t 2 + 64t
•
4: h4 = !16t 2 + 80t
5. Using the function rules for the four graphs, predict the rules for the 5th and 6th
graphs that were described in (3). To check your predictions, verify the values
that you found for the vertices of these graphs using your new function rules.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.