Whatcom Math Project
How High Can You Throw a Tennis Ball?
Course: Algebra 2
Grade Level: 9 − 12
College Readiness Standard Name(s) and Number(s):
Standard 8: Functions
The student accurately describes and applies function
concepts and procedures to understand mathematical
relationships.
Standard 1: Reasoning and Problem-solving
The student uses logical reasoning and mathematical
knowledge to define and solve problems.
Standard 2: Communication
The student can interpret and communicate mathematical knowledge and
relationships in both mathematical and everyday language.
Standard 3: Connections
The student extends mathematical thinking across mathematical content areas,
and to other disciplines and real life situations.
Student Attributes:
Perseveres when faced with time-consuming or complex tasks.
Pays attention to detail.
Student Learning Outcomes − College Readiness Standard Component and
Number:
8.2 Represent quadratic functions using and translating among words, tables,
graphs, and symbols.
8.3 Analyze and interpret features of a quadratic function.
8.4 Model situations and relationships using quadratic functions.
1.1 Analyze a situation and describe the problem(s) to be solved.
1.2 Formulate a plan for solving the problem.
2.2 Use symbols, diagrams, graphs, and words to clearly communicate mathematical
ideas, reasoning, and their implications.
3.1 Use mathematical ideas and strategies to analyze relationships within mathematics
and in other disciplines and real life situations
1
Learning Objectives:
1. Students will be able to understand the quadratic equation for an object traveling
perpendicular to the ground.
2. Students will be able to create a quadratic equation for a ball thrown straight up
in the air given the length of time it takes a ball to land after it is thrown.
3. Students will be able to calculate the maximum height of the ball from their
equation.
4. Students will be able to graph their equation by hand and on their graphing
calculator.
Prerequisite Skills:
Students display understanding of quadratic equations/functions.
Students can build a quadratic equation given a point and/or the vertex.
Students can calculate the vertex of a quadratic function.
Students can graph a quadratic function on their calculator and find the vertex.
Students can substitute values in for variables and solve for different unknowns.
Material for Students:
Handout (included with the lesson)
Tennis Ball
Meter stick
Stopwatch
TI-Graphing calculator
Teaching Aids:
Large, flat area like the infield of a track or soccer field
Document camera (optional)
Graphing Calculator Presenter (optional)
2
Estimated Time For Completion:
Pre-Assessment: 20 minutes to administer and discuss with the students (completed
prior to the day of the lesson)
Lesson: 1 block period or 2 shorter periods
Introduction: 15 minutes
Activity: 45 minutes
Classwork: 20 minutes
Closure: 5 minutes
Extension: 10 minutes
Post-Assessment: 20 minutes to discuss the results from the experiment and share
student solutions. Can be completed at the end of the day or the next day.
References:
None
3
I.
PREASSESSMENT:
Name: _________________________
1 point per question
Students display understanding of quadratic equations/functions
1. Identify the quadratic function.
a. y = 3x + 4
b. y = 2x2 + 5x − 2
c. y = 5x3 + 4x2 + x + 3
d. y = −32 + 4x + 4
2. Identify the quadratic function that is written in vertex/graphing form.
a. y = 4(x + 2) – 5
b. y = 4x2 + 3x – 5
c. y = 2(x + 4)3 – 5
d. y = 3(x – 2)2 + 1
3. If a quadratic is used to model the height of a ball thrown straight up in the air as
a function of time, where in the graph would the maximum height of the ball be
found?
a. At the y-intercept
b. At the x-intercepts
c. At the vertex
d. When the time is zero
Students can graph a quadratic function on their calculator and find the vertex
4. Plot y = x2 − 10x + 16 on your graphing calculator. Identify the vertex.
a. (5, 9)
b. (5, −9)
c. (0, 16)
d. (2,0) & (8,0)
4
2 points per question
Students can create a quadratic equation given a point and/or the vertex by hand
5. You are given the graph of a parabola with a vertex of (5, 10) and passing
through the point (2, 1). Create a quadratic equation in vertex form by hand and
showing all work.
Students can calculate the vertex of a quadratic function
6. Calculate the vertex of y = −16x2 + 80x by hand and showing all work.
Students can substitute values in for variables and solve for different unknowns
7. If y = x2 + 5x + 4 and y = 10, what would x equal? Show all work.
8. The standard form of a quadratic equation is y = ax2 + bx + c (a ≠ 0). Create a
quadratic equation in standard form if a = −16, the y-intercept was (0, 4) and it
passed through the point (5, 404). Show all work.
5
II.
PRE−ASSESSMENT SOLUTIONS
1 point per question
Students display understanding of quadratic equations/functions
1. Identify the quadratic function
a. y = 3x + 4
(linear function)
b. y = 2x2 + 5x – 2
c. y = 5x3 + 4x2 + x + 3(cubic function – highest power is 3)
d. y = −32 + 4x + 4
(linear function since a constant is being squared)
2. Identify the quadratic function that is written in vertex/graphing form
a. y = 4(x + 2) – 5
(linear function)
b. y = 4x2 + 3x – 5
(quadratic, but in standard form)
c. y = 2(x + 4)3 – 5
(cubic)
d. y = 3(x – 2)2 + 1
3. If a quadratic is used to model the height of a ball thrown straight up in the air as
a function of time, where in the graph would the maximum height of the ball be
found?
a. At the y-intercept
(height at time zero)
b. At the x-intercepts
(time when y = 0)
c. At the vertex
d. When the time is zero
(y-intercept/initial height)
Students can graph a quadratic function on their calculator and find the vertex
4. Plot y = x2 − 10x + 16 on your graphing calculator. Identify the vertex.
a. (5, 9) (plugged −10 into calculator as +10)
b. (5, −9)
c. (0, 16) (y−intercept)
d. (2,0) & (8,0) (x−intercepts)
6
2 points per question
Students can create a quadratic equation given points and/or the vertex by hand
5. You are given the graph of a parabola with a vertex of (5, 10) and passing
through the point (2, 1). Create a quadratic equation in vertex form by hand and
showing all work.
2−point response: Students should begin with the vertex form of a quadratic
equation, y = a(x – h)2 + k, plug in the vertex (h, k) and point (x, y) and solve for
the “a” value. In this case, a = −1. The correct answer is y = −(x – 5)2 + 10.
1−point response: Student does everything for the 2-point response, but
incorrectly solves for the value of a.
0−point response: Shows little to no understanding or makes more than one
mistake.
Students can calculate the vertex of a quadratic function
6. Calculate the vertex of y = −16x2 + 80x by hand and showing all work.
2−point response: Students should you x = −b/2a to determine the x−value
(2.5) of the vertex and then plug this in to find the y-value (100). The correct
answer is (2.5, 100).
1−point response: Student does everything for the 2-point response, but
incorrectly solves for one or both variables.
0−point response: Shows little to no understanding or makes more than one
mistake.
Students can substitute values in for variables and solve for different unknowns
7. If y = x2 + 5x + 4 and y = 10, what would x equal? Show all work.
2−point response: Students should plug 10 in for y and solve for x by setting
equal to zero and solving algebraically by the method of their choice (factoring,
the quadratic formula, or completing the square). The correct answer is x = 1 or
−6.
1−point response: Student does everything for the 2-point response, but only
solves for one of the two answers or makes a simple error, but has two answers.
0−point response: Shows little to no understanding or makes more than one
mistake.
7
8. The standard form of a quadratic equation is y = ax2 + bx + c (a ≠ 0). Create a
quadratic equation in standard form if a = −16, the y-intercept was (0, 4) and it
passed through the point (5, 404). Show all work.
2−point response: Students should plug in the given values and solve for b
which is 160. The correct answer is y = −16x2 + 160x + 4.
1−point response: Student does everything for the 2-point response, but has an
incorrect solution for b.
0−point response: Shows little to no understanding or makes more than one
mistake.
8
III.
INTRODUCTION:
This experiment summarizes and extends the skills students have gained during a unit
on quadratics.

If you have not done so already, begin the first day going over the preassessment questions from the previous day. Clarify any student questions prior
to beginning the lesson.
After reviewing the pre-assessment, begin the lesson as follows:

Ask the class how high they think they could throw a tennis ball if they threw it
straight up in the air.
Possible answers: 20, 30, 40 feet, maybe 100 feet, the comedian will say 1 mile

After the students have made a guess, have them brainstorm how they could
actually find it out.
Possible ideas:
 Use an electronic measuring device
 Time how fast it’s flying and how long it’s in the air
 The time it’s in the air
 Use a known height (on a wall) and throw the ball next to it
 Students strong in Geometry might think of using angles and time it took the
ball to travel between them
The idea is to get the students to come up with lots of ideas that are generally difficult to
actually do or buy. After the students have brainstormed, tell them there is an easier
way to find the maximum height without actually measuring the height.

After they have thought about how to measure the height, ask them what factors
affect the height of the ball?
Possible examples:
 Gravity (the same no matter where you are on the Earth)
 How hard/fast the ball is thrown (the initial velocity)
 The starting height (the initial height)
 Wind (may affect the time slightly, but not significant)
 Angle the ball is thrown (this will affect their maximum height – encourage the
students to throw the ball straight up. I had a long discussion with our physics
teacher about this and although it would affect the time in the air, which we
use to calculate the maximum height, the angle does not matter)
 Where it lands (especially if it lands lower than the initial height)
9

After they have thought about what affects the height, ask them about the
graph… “Given that the height is a function of time, what type of graph might you
use to model this function?”
Possible examples:
 Linear (incorrect because the ball can’t rise infinitely and it assumes that the
ball rises at a constant rate)
 Absolute value (incorrect because although it does have a maximum height
(vertex) it still requires the ball to ascend and descend at a constant rate)
 Quadratic (correct because the ball will rise quickly, slow down, reach a
maximum height, slowly begin descending, but increase in speed as it
approaches the ground. If quadratic is given at first, ask them about the other
functions and why the students wouldn’t use them.

Once the students have agreed that a quadratic function is the best model, ask
them how will we calculate the maximum height?
Possible examples:
 Average the x-intercepts to find the axis/line of symmetry (x-value of the
vertex) and plug that in to find the y-value (height) – but how do we figure out
the x-intercepts? My students incorrectly think of the first one being at the
origin instead of realizing the initial height affects this,
 Plug the equation into their graphing calculator and look at the graph or table
to find it, calculate the vertex algebraically using x = −b/2a (this is the best
example and the most straight forward since it always works no matter how
complicated the quadratic equation.
 Other methods would work, factoring, x-intercepts, completing the square, but
these methods are cumbersome or not possible. In addition, I require
students to do their work by hand first and verify it on their calculator)

“Since we know how to find the maximum height when you are given the
equation, what’ missing?” Answer: the equation… but how are we going to create
it?
Possible examples:
 The y-intercept is the initial height of the ball or the height when it is first
thrown;
 Students may know how to build an quadratic equation given three points, but
this would not be feasible since we cannot determine enough heights at
certain times;
 Students may realize that the equation must begin with a negative in order for
it to open down or that they can plug in a height and time.
10
IV.

LESSON:
Introduce the quadratic function relating height and time: h(t) = −16t2 + v0t + h0
with h(t) representing height (as a function of time) in feet, t representing time in
seconds, v0 representing the initial velocity in feet per second, h0 representing
the initial height in feet, and −16 being the gravity constant in feet per second
squared.
This equation is a standard when modeling an object fired/thrown straight up in the air.
Reiterate that height is a function of the time in the air: h(t).

Ask the students what we need to find out in order to create a workable equation
(v0 and h0).
The students should respond that we can measure the initial height – it is the height
when the ball is released by the thrower. How would we measure it? Have the students
take meter sticks to measure the height of the students hand when they anticipate the
ball will be released. You may encourage the students to try this a few times to see
what height the thrower releases the ball at. Also, what if the student will release it at a
height of 5 feet 9 inches? Students may need to be instructed on how to convert their
height into only feet (5 feet, 9 inches = 5.75 feet).
The harder variable to find is the initial velocity. Students may say that we could use a
radar gun, but my students don’t have one and even if you did, it would be hard to
accurately determine it. Students may also say that we need a height and
corresponding time. These will be helpful for the next part of the lesson, but students
should understand that a quadratic function written this way has an infinite number of
heights and time that exist and these are the two variables that should remain variables
in the final equation.

Since we have determined that we need the initial velocity to create the initial
equation, how can we find it without a way of accurately measuring it?
Focus the students’ attention on the fact that after plugging the initial height into the
equation there are three variables left: the initial velocity, height and time. If we can
determine a height and corresponding time, we can plug those in and solve for the
initial velocity. See example below:

“Now that we have figured out that we need a height and time, how are we going
to find one? Have the students brainstorm how we can find an accurate height
and time.
11
Possible answers:
 Use the initial height and time zero (plugging this time in will cause all the
other variables will cancel out);
 We could time the ball until it reaches the initial height again (we’d have to
catch it, but this is not easy or accurate);
 We could create a net or catch the ball at a certain height and stop the watch
when it is caught (not easy to catch and/or build something);
 Eventually you will want to lead the students to stopping the watch when the
ball hits the ground.

So, all of this was just to create the equation. How do we determine the
maximum height?
Students should know how to calculate the vertex from a quadratic equation as
mentioned earlier in the lesson.

Ask the students how accurate they think this will be? What are some of the
factors that might affect its accuracy? How can we make it more accurate?
The students should realize that doing an experiment multiple times will help it be more
accurate. Tell them to throw the ball a minimum of three times for each person. They
should take the average of their throws and use this time to build their own equation.

Examples:
Student A releases the ball at a height of 6 feet, 3 inches (6.25 feet). After three
trials, the following times are recorded when the ball hits the ground: 3.1, 3.5, 3
seconds. The ball hits the ground, on average, after 3.2 seconds.
Begin with the equation:
Insert known values:
Solve for v0:
h(t) = −16t2 + v0t + h0
0 = −16(3.2)2 + v0(3.2) + 6.25
0 = −163.84 + 3.2v0 + 6.25
157.59 = 3.2v0
49.246875 = v0
Equation for Student A:
h(t) = −16t2 + 49.246875t + 6.25
Calculate the vertex:
t = −b/(2a)
t = −49.246875/(2*−16)
t = −49.246875/−32
t = 1.539023438… seconds (the time of the vertex)
h(1.53…) = −16(1.53…)2 + 49.246875(1.53…) + 6.25
h(1.53…) = 44.1446…feet (the height of the vertex)
Student A’s ball had a maximum height of 44.145 feet high (after 1.539 seconds).
Their throw was 38.145 feet high after subtracting their initial height.
12
Student B releases the ball at a height of 6 feet. After three trials, the following
times are recorded when the ball hits the ground: 4.5, 4.7, 4.4 seconds. The ball
hits the ground, on average, after approximately 4.53 seconds.
Begin with the equation:
Insert known values:
Solve for v0:
h(t) = −16t2 + v0t + h0
0 = −16(4.53)2 + v0(4.53) + 6
0 = −328.82 + 4.53v0 + 6
322.82 = 4.53v0
71.21 = v0
Equation for Student B:
h(t) = −16t2 + 71.21t + 6
Calculate the vertex:
t = −b/(2a)
t = −71.21/(2*−16)
t = −41.21/−32
t = 2.23 seconds (the time of the vertex)
h(2.23) = −16(2.23)2 + 71.21(2.23) + 6
h(2.23) = 85.23feet (the height of the vertex)
Student B’s ball had a maximum height of 85.23feet high (after 2.23 seconds).
Their throw was 79.23 feet high after subtracting their height.
Student C releases the ball at a height of 5 feet, 3 inches (5.25 feet). After three
trials, the following times are recorded when the ball hits the ground: 1.6, 1.6, 1.9
seconds. The ball hits the ground, on average, after 1.7 seconds.
Begin with the equation:
Insert known values:
Solve for v0:
h(t) = −16t2 + v0t + h0
0 = −16(1.7)2 + v0(1.7) + 5.25
0 = −46.24 + 1.7v0 + 5.25
40.99 = 1.7v0
24.11 = v0
Final equation for student: h(t) = −16t2 + 24.11t + 5.25
Calculate the vertex:
t = −b/(2a)
t = −24.11/(2*−16)
t = −24.11/−32
t = 0.75 seconds (the time of the vertex)
h(0.75) = −16(0.75)2 + 24.11(0.75) + 5.25
h(0.75) = 14.34feet (the height of the vertex)
Student C’s ball had a maximum height of 14.34feet high (after 0.75 seconds).
Their throw was 9.09 feet high after subtracting their height.
13
V.
APPLICATION:
This part of the lesson will require students to work in teams of 4 students; smaller
teams are feasible if you have enough supplies.
Each group will need the handout provided with the lesson, a tennis ball, a stopwatch, a
meter stick and a graphing calculator. Groups should be self-paced with the teacher
monitoring appropriate behavior and helping with any student difficulties.

Follow the introduction as outlined above.

Introduce the quadratic function relating height and time: h(t) = −16t2 + v0t + h0
as outlined in the lesson section above.

Student groups will go outside and perform the experiment. Allow enough time
for each student to throw the ball. This will allow them the opportunity to build an
equation unique to them. If there are enough stop watches, multiple students
could time each flight in order to get a more accurate time.

Before any group begins, model the experiment with one of the groups:
o Have the student approximate the height when they will release the ball.
o Have another students measure this height – This is the initial height of
the ball.
o When a student is throwing, start the watch at the point when the ball is
released from the hand (initial height).
o Stop the watch at the instant the ball touches the ground – This will give
them a height, h(t), and a corresponding time, t.
o Since the time pertains to the thrower (so that they may build their own
equation), have the students record for each other.
o Encourage the students to practice throwing at least once before
conducting their trials.

After completing the experiment outside, students should return to the classroom
in order to work on building their equation.
o Students should begin with the basic quadratic equation relating height
and time: h(t) = −16t2 + v0t + h0
o Students should plug in what they know: h0 is their initial height, h(t) is the
height when they stopped the watch, 0 feet, and t is the time when the ball
hit the ground (their average of the three times they recorded).
o Students should then solve for the initial velocity, v0
o This will allow them to build a quadratic to measure any height at any time
based on their data.
14

Once the students have a workable quadratic and have them check it for
accuracy (by hand or on their graphing calculator).
o Students can check by hand by plugging in zero for the height and the
time it reached the ground.
o Students can check with their graphing calculator by plugging the equation
in and checking the point when the ball hit the ground.

After the students have checked their equations, have them calculate the
maximum height (find the vertex) by hand and with their calculator.
o Students can find the vertex by hand by solving t = −b/2a and plugging
this result in for t in the equation and simplifying to find the corresponding
h(t). Refer to the example in Section IV (Lesson).

Once all of the students have found their heights, have them determine who
threw the ball the highest in their group. Have everyone share their data with the
class and talk about the results...
o Which student threw the ball the highest?
o In order to find the actual distance the ball was thrown, have the students
subtract their initial height from their maximum height. Did the same
student still throw the ball the highest?
EXTENSION:
This extension could be used for classes where stoichiometry/dimensional analysis/unit
conversions are covered or would like to be covered. An example is shown below and
uses the same data as the example in Section IV (Lesson)

Have the students use stoichiometry/dimensional analysis/unit conversions in
order to convert their initial velocity from feet per second to miles per hour.
Which student threw the ball the fastest?
Example:
Student A threw the ball will an initial velocity of 49.246875 feet/sec…
49.246875feet/sec ● 60sec/1min ● 60min/1hour ● 1mile/5280feet = 33.577mph
CONCLUSION:
The students’ initial velocity should be between 20 and 80 feet/sec. Their maximum
heights should be between 10 and 100 feet high.
15
VI.
RESOURCES:
HOW HIGH CAN YOU THROW A BALL?
Student Resource Packet
TEACHER:
NAME:
DATE:
PERIOD:
1.
Explain why the height as a function of time is best represented by a quadratic
equation.
2.
The basic equation for an object traveling perpendicular to the earth and being
affected by gravity is h(t) = −16t2 + v0t + h0 with h(t) representing height (as a
function of time) in feet, t representing time in seconds, v0 representing the initial
velocity in feet per second, h0 representing the initial height in feet, and −16 being
the gravity constant in feet per second squared.
a. Of the four variables, which can you determine or establish at the start of the
experiment? How can we determine them?
b. Of the four variables, which can you determine during the experiment? How
will you determine it?
c. Of the four variables, which is the most difficult to determine? How will you
determine it?
16
3.
Perform the experiment a minimum of three times in order to determine the values
needed for part 2b. Record that information here:
Trial #
1
2
3
Time (t)
4.
Find the average of your three throws.
5.
What values do you now have for your equation? Plug them in and solve for the
remaining variable in order to create a quadratic equation that you can use to
model the height of you ball as a function of time.
Your Equation:
_____________________________________
6.
Check the accuracy of your equation by hand or using your calculator. Show your
work or explain how your calculator verified your solution.
7.
Now that you have a working quadratic to model the height of your ball at any
given time, use it to calculate the maximum height.
17
8.
Graph your function by plotting a point every .2 seconds (Use increments of .2
seconds on the t-axis, but you determine the best fit for the h(t)-axis). In addition,
label the vertex, x-intercept, and y-intercept.
9.
Determine the distance your threw the ball (max height minus the initial height)
Extension problem…
10. Your initial velocity was in feet per second. Determine the speed of your ball in
miles per hour.
18
VII. ASSESSMENT:
Multiple Choice – 1 Point Each
1. Identify the quadratic function.
a. y = x2 − 4x + 1
b. y = 2x3 − 3x2 + 2x + 1
c. y = −x2 + 4x3 + 4
d. y = 3x + 4
2. Identify the quadratic function that is written in vertex/graphing form.
a. y = 2x2 + 5x + 1
b. y = −4x – 5
c. y = −(x − 3)3 + 4
d. y = (x + 3)2 – 5
3. Identify the vertex of y = 2(x + 5)2 − 2.
a. (0, 48)
b. (5, −2)
c. (−2, −5)
d. (−5, −2)
4. Identify the vertex of y = x2 + 14x + 24.
a. (0, 24)
b. (−2, 0)
c. (−7, −25)
d. (−7, −123)
19
5. Choose the quadratic equation that could represent height as a function of time
for a rocket fired into the air.
a. h(t) = −16t2 + 160t
b. h(t) = −t2 + 80t + 2
c. h(t) = 16t2 + 100t + 2
d. h(t) = −5t2 + 60t + 1
6. Identify the situation that would best be modeled by a quadratic function.
a. The population of the earth as a function of time since the year 1000.
b. The height of a punted football as a function of time.
c. The amount of water in a bucket that has a hole in it as a function of time.
d. The profit made as a function of the number of $5 items sold.
7. Given y = x2 + 6x − 16, identify the y-intercept.
a. (0, −16)
b. (−16, 0)
c. (−3, −7)
d. (0, −8) & (0, 2)
8. Identify the equation that has a vertex at (2, 4) and y-intercept of 40.
a. y = 2x2 + 4x + 1
b. y = x2 − 4x + 40
c. y = 9(x − 2)2 + 4
d. y = (x + 2)2 + 4
20
9. Given h(t) = −16t2 + 160t, calculate when h(t) = 0.
a. t = 0
b. t = 5
c. t = 10
d. t = −10
Short Answer – 2 Points Each
10. A rocket that takes off at a velocity of 200ft/sec from a height of 2 feet. The
constant for gravity is −16 feet/second squared. Create a quadratic equation to
model the height as a function of time.
11. A ball is thrown in the air and the equation h(t) = −16t2 + 160t + 8 is used to
model its height as a function of time. Show all work necessary to calculate the
maximum height the ball achieves by hand.
21
12. A ball is thrown upwards from a height of 6 feet and lands on the ground 3.22
seconds later. The constant for gravity is −16 feet/second squared. Write an
equation to model the height of the ball as a function of time. This should be
done algebraically by hand and showing all work.
13. A ball is thrown in the air and the equation h(t) = −16t2 + 160t + 8 is used to
model its height as a function of time. Calculate the time when the ball hits the
ground by hand and showing all work. Decimal approximations should be
rounded to the nearest hundredth.
22
Extended Response − 4 points each
14. A rocket takes off from a launch platform and the following data points are
recorded.
t
h(t)
0
12
1
236
2
428
3
588
4
716
5
812
6
876
7
908
8
908
9
876
10
812
11
716
12
588
13
428
14
236
15
12
a. Analyze the data and explain why the data would best be represented as
a quadratic function.
b. Create a quadratic equation to model the height as a function of time.
c. Use your equation to calculate the vertex. Does is concur with your data?
d. Explain the significance in the point (0,12) and how it also verifies whether
or not your equation is correct.
23
15. In order to make more money, a company has decided to raise their selling price
for a product in $5 increments. They also realize that they will sell 50 fewer
products for each $5 increase. A mathematician in the company created the
equation p(n) = (10 + 5n)(500 – 50n) to model the profit they would make where
p(n) is the total profit and n is the number of $5 price increases. Be sure to show
all work for
a. Simplify the equation to show that it is a quadratic.
b. Use the equation from part a. to calculate the initial value of the product.
c. Use the equation from part a. to calculate the number price increases that
will result in no profit for the company.
d. Use the equation from part a. to calculate the maximum price they should
charge for their product and the number of price increases necessary to
achieve it.
e. Create a graph to show the profit as a function of the number of increases
in price (scale the x-axis increments by 1 and the y-axis increments by
$500). Does your graph verify your calculations?
24
VIII. ASSESSMENT SOLUTIONS:
Multiple Choice – 1 Point Each
1. Identify the quadratic function.
CRS 8.2b: Describe the algebraic features of a function and the features of its
graph.
a. y = x2 − 4x + 1
b. y = 2x3 − 3x2 + 2x + 1
(cubic)
c. y = −x2 + 4x3 + 4
(cubic, not in descending order)
d. y = 3x + 4
(linear)
2. Identify the quadratic function that is written in vertex/graphing form.
CRS 8.2b: Describe the algebraic features of a function and the features of its
graph.
a. y = 2x2 + 5x + 1
(standard form)
b. y = −4x – 5
(linear)
c. y = −(x − 3)3 + 4
(cubic)
d. y = (x + 3)2 – 5
3. Identify the vertex of y = 2(x + 5)2 − 2.
CRS 8.3c: Identify the extrema of a quadratic function.
a. (0, 48)
(y-intercept)
b. (5, −2)
(x-value is not opposite)
c. (−2, −5)
(x & y are switched)
d. (−5, −2)
25
4. Identify the vertex of y = x2 + 14x + 24.
CRS 8.3c: Identify the extrema of a quadratic function.
a. (0, 24)
(y-intercept)
b. (−2, 0)
(x−intecept)
c. (−7, −25)
d. (−7, −123)
(incorrectly multiplied x = −7… (−7)2 ≠ −49)
5. Choose the quadratic equation that could represent height as a function of time
for a rocket fired into the air.
8.4a: Choose a function suitable for modeling a real word situation.
a. h(t) = −16t2 + 160t
b. h(t) = −t2 + 80t + 2
(needs the gravity constant of −16)
c. h(t) = 16t2 + 100t + 2
(the gravity constant is not −16)
d. h(t) = −5t2 + 60t + 1
(needs the gravity constant of −16)
6. Identify the situation that would best be modeled by a quadratic function.
CRS 8.4a: Choose a function suitable for modeling a real word situation
a. The population of the earth as a function of time since the year 1000.
(exponential)
b. The height of a football kicked into the air as a function of time.
c. The amount of water in a bucket that has a hole in it as a function of time.
(linear)
d. The profit made as a function of the number of $5 items sold.
(linear)
26
7. Given y = x2 + 6x − 16, identify the y-intercept.
CRS 8.3b: Identify the y-intercept of a quadratic function.
a. (0, −16)
b. (−16, 0)
(put the y value in the x location)
c. (−3, −7)
(vertex)
d. (0, −8) & (0, 2)
(x-intercept)
8. Identify the equation that has a vertex at (2, 4) and y-intercept of 40.
CRS 8.4c: Abstract mathematical models from word problems.
a. y = 2x2 + 4x + 1
(the y-intercept is 1)
b. y = x2 − 4x + 40
(y-intercept is 40, but vertex is incorrect)
c. y = 9(x − 2)2 + 4
d. y = (x + 2)2 + 4
(vertex is correct, but y-intercept is incorrect)
9. Given h(t) = −16t2 + 160t, calculate when h(t) = 0.
CRS 8.3b: Identify the zeros of a quadratic function.
a. t = 0
(one of the two solutions)
b. t = 5
(time at the vertex)
c. t = 10
(one of the two solutions)
d. t = 0 or 10
27
Short Answer – 2 Points Each
10. A rocket that takes off at a velocity of 200ft/sec from a height of 2 feet. The
constant for gravity is −16 feet/second squared. Create a quadratic equation to
model the height as a function of time.
CRS 8.4c: Abstract a mathematical model from a word problem.
2−point response: Students show understanding of mathematical models by
completing each of the following:
 Substituting −16 for a, 200 for b and 2 for c and making the equation h(t) =
−16t2 + 200t + 2.
 Writes h(t) = −16t2 + 200t + 2.
1−point response: Students show some understanding of mathematical models
by completing one of the following:
 Substituting −16 for a, 200 for b and 2 for c and making the equation h(t) =
−16t2 + 200t + 2.
 Writes h(t) = −16t2 + 200t + 2.
0−point response: Shows little to no understanding or makes more than one
mistake.
11. A ball is thrown in the air and the equation h(t) = −16t2 + 160t + 8 is used to
model its height as a function of time. Calculate the maximum height the ball
achieves by hand and showing all work.
CRS 8.3c: Identify extrema (vertex) of a quadratic equation.
2−point response: Students show how to identify the extrama of a quadratic
equation by completing the following:
 should use t = −b/2a (or another suitable method like factoring or
completing the square) to find the time of the vertex and then plug this
time in to find the height.
 The maximum height is 408 feet after 5 seconds.
1−point response: Students show how to identify the extrama of a quadratic
equation by completing the following:
 should use t = −b/2a (or another suitable method like factoring or
completing the square) to find the time of the vertex and then plug this
time in to find the height.
 Student makes an error in calculating the time and achieves the wrong
height.
0−point response: Shows little to no understanding of solving quadratics.
28
12. A ball is thrown upwards from a height of 6 feet and lands on the ground 3.22
seconds later. The constant for gravity is −16 feet/second squared. Write an
equation to model the height of the ball as a function of time. This should be
done algebraically by hand and showing all work.
CRS 8.4c: Abstract a mathematical model from a word problem.
2−point response: Students show understanding of how to abstract a
mathematical model from a word problem by completing the following:
 The y-intercept 6 and the point (3.22, 0) are given. Students should
substitute these values in and solve for b.
 Writes h(t) = −16t2 + 49.66t + 6.
1−point response: Students show some understanding of how to abstract a
mathematical model from a word problem by completing the following:
 The y-intercept 6 and the point (3.22, 0) are given. Students should
substitute these values in and solve for b.
 Writes an incorrect equation like h(t) = −16t2 + 26t + 6.
0−point response: Students shows little or no understanding of how to abstract
a mathematical model from a word problem.
13. A ball is thrown in the air and the equation h(t) = −16t2 + 160t + 8 is used to
model its height as a function of time. Calculate the time when the ball hits the
ground by hand and showing all work. Decimal approximations should be
rounded to the nearest hundredth.
CRS 8.4b: Determine and interpret the meaning of zeros.
2−point response: Students show how to determine and interpret the meaning
of zeros by completing the following:
 Substituting 0 in for the h(t) and solve using the quadratic by hand using
the method of their choice (quadratic formula, factoring, completing the
square).
 Write t = 0.05sec and 10.05sec
1−point response: Students show how to determine and interpret the meaning
of zeros by completing the following:
 Students should substitute 0 in for the h(t) and solve using the quadratic
by hand using the method of their choice (quadratic formula, factoring,
completing the square).
 The formula should be simplified correctly although the answer will be
incorrect.
 Student should still write t = two answers.
0−point response: Students show little or no ability to determine and interpret
the meaning of zeros by completing the following:
29
Extended Response − 4 points each
14. A rocket takes off from a launch platform and the following data points are
recorded.
t
h(t)
0
12
1
236
2
428
3
588
4
716
5
812
6
876
7
908
8
908
9
876
10
812
11
716
12
588
13
428
14
236
15
12
a. Analyze the data and explain why the data would best be represented as
a quadratic function.
b. By hand, create a quadratic equation to model the height as a function of
time in either vertex or standard form.
c. Use your equation to calculate the vertex. Does is concur with your data?
d. Explain the significance in the point (0,12) and how it also verifies whether
or not your equation is correct.
CRS 1.1a Extract necessary facts and relationships from given information.
CRS 8.3c: Identify extrema (vertex) of a quadratic equation.
CRS 8.4c: Abstract a mathematical model from a word problem.
4−point response:
Students show understanding of extracting necessary facts, identifying extrema
and abstracting mathematical models by completing all of the following:
14a. The graph has a vertex between 7 and 8 seconds
The points are mirrored
The rate of change increases rapidly at first, slows near the vertex, is zero
at the vertex, slowly begins descending and descends more rapidly as
time increases.
14b. Students should show their work and write h(t) = −16t2 + 240t + 12
14c. Students should show their work and write t = 7.5 seconds and 912 feet
14d. Students should explain that (0, 12) is the y-intercept and can be verified
by substituting 0 in for t and solving (h(t) will equal 12) or by substituting
12 in for h(t) and solving for t (t will equal both 0 and 15)
3−point response:
3 of the 4 parts are done correctly
2−point response:
2 of the 4 parts are done correctly
1−point response:
1 of the 4 parts are done correctly
0−point response: Shows little to no understanding of quadratics or writing
quadratic equations.
30
15. In order to make more money, a company has decided to raise their selling price
for a product in $5 increments. They also realize that they will sell 50 fewer
products for each $5 increase. A mathematician in the company created the
equation p(n) = (10 + 5n)(500 – 50n) to model the profit they would make where
p(n) is the total profit and n is the number of $5 price increases. Be sure to show
all work for
a. Simplify the equation to show that it is a quadratic.
b. Use the equation from part a. to calculate the initial value of the product.
c. Use the equation from part a. to calculate the number price increases that
will result in no profit for the company.
d. Use the equation from part a. to calculate the maximum price they should
charge for their product and the number of price increases necessary to
achieve it.
e. Create a graph to show the profit as a function of the number of increases
in price (scale the x-axis increments by 1 and the y-axis increments by
$500). Does your graph verify your calculations?
CRS 1.2b
Choose concepts, strategies, representations, models, and tool
well-suited to solving the problem.
CRS 8.4b: Determine and interpret the meaning of intercepts zeros and
extrema.
The remainder of the solutions are on the following page
31
4−point response:
Students show understanding of concepts, strategies, representations, models,
and tool well-suited to solving the problem and determining and interpreting the
meaning of intercepts zeros by completing all of the following:
15a.
15b.
15c.
15d.
15e.
Student writes p(n) = −250n2 + 2000n + 5000
Students should show their work, substitute n = 0 and write the initial value
is $5000
Students should show their work, substitute p(n) = 0 and write n = 10
prices increases will result in p(n) = 0 (the value of n = −2 should be
ignored).
Students should show their work, use n = −b/2a or another method to
algebraically calculate the maximum of $9000 after 4 price increases.
Graph should look like the graph on the previous page.
3−point response:
4 of the 5 parts are done correctly
2−point response:
3 of the 5 parts are done correctly
1−point response:
21 of the 5 parts are done correctly
0−point response: Shows little to no understanding of quadratics or writing
quadratic equations.
32
IX.
EXTENSIONS:
Students could watch the movie “October Sky” and see how the boys used a similar
technique to find out how high their rockets were flying before or after the experiment.
Students could also build their own water rockets or model rockets and use the same
technique calculate the maximum height of the rocket.
33