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Bridge to
College
Mathematics
Unit 6. Quadratic Functions
Student Manual
Name
1
Student Manual
Bridge to College Mathematics . Unit 6. Student Manual
Unit 6. Quadratic Functions
Table of Contents
Lesson 1.............................................................................................................3
Lesson 2.............................................................................................................5
Lesson 3.............................................................................................................8
Lesson 4...........................................................................................................17
Lesson 5...........................................................................................................21
Lesson 6...........................................................................................................25
Lesson 7...........................................................................................................36
Lesson 8...........................................................................................................38
Lesson 9...........................................................................................................42
Lesson 10.........................................................................................................45
Lesson 11.........................................................................................................48
Lesson 12.........................................................................................................51
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 1
Task #1: Quadratic or Not?
In your groups, use the illustrations to help you in defining key features of quadratic graphs.
Prepare a toolkit to share with the class.
The following are graphs of quadratic functions:
3
5
4
3
2
1
2
1
-3
-2
-1
0
-3
1
-2
4
3
2
-1 -1
-2
-3
-4
1
2
3
1
-2
-1
0
1
2
The following are not graphs of quadratic functions:
8
7
6
4
3
2
1.0
6
0.7
2
-2
-1
0
0.5
1
0.3
2
0.1
-6
-3
-2
-1
0
1
2
3
Describe how graphs of quadratic functions differ from non-quadratic function graphs, and give
characteristics of quadratic function graphs. Describe symmetries, asymptotes and concavity; discuss
domain and range; comment on how the graph decreases and/or increases. Include discussion on
rates of change.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 1
Quadratics
Job Descriptor Cards
Reading Manager
Spying Monitor
• Reads ALL parts of the assignment
and problems out loud to the group
(others follow along).
• Monitors group progress relative
to other groups.
• Ensures group members understand
assignments.
• Keeps group focused on the task(s).
• Only member in group that can talk/ask
questions outside of group.
Quality Controller
Recording Time Keeper
• Ensures that all group members
can EXPLAIN and JUSTIFY each
response (random checks occur by
management).
• Keeps track of time.
• Makes sure members are completing
ALL problems in appropriate notebook.
• Keeps group supplies organized and
neat.
• Reports missing items.
4
• Checks in with other groups for
comparison.
• When asked, shares group responses.
• Responsible for ensuring “public record” (posting of answers, group posters, etc.) is completed.
Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 2
Task #2: Coefficients and Graphs
Graph some or all of the functions below, or create your own functions to graph, in order to answer the
three questions given. Your answer to each question should include references to and sketches of
several graphs.
f(x) = x2 + 2x – 3
g(x) = -x2 + 2x – 3
h(x) = 3x2 + 2x – 3
k(x) = -3x2 + 2x – 3
m(x) = x2 + 2x + 3
n(x) = -x2 + 2x + 3
p(x) = -9x2 + 2x + 4
q(x) = 5x2 + 2x +5
(1) How does the value of c relate to the graph of the function?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 2
(2) How can you tell if the graph will open up or down?
(3) What determines the "width" of the parabola?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 2
Task #3: Coefficients and Projectile Motion
In our projectile motion example, we had a function: h(t) = -16t2 + 5t + 15
1. Where did each number (that is, -16, 5, and 15) come from?
2. Which values are free to change? Which are not?
3. Consider what would happen to the height graph of the object if you did change the values that
are free to change. Would it have a different starting point? Have a different maximum height?
Be in the air for a longer or shorter time? Sketch at least three example of graphs for each
number that can change to illustrate your points.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
Task #4: Egg Launch Contest
NAME:
DATE:
Mr. Rhodes’ class is holding an egg launching contest on the football field. Teams of students have
built catapults that will hurl an egg down the field. Ms. Monroe’s class will judge the contest. They
have various tools and ideas for measuring each launch and how to determine which team wins.
Team A used their catapult and hurled an egg down the football field. Students used a motion
detector to collect data while the egg was in the air. They came up with the table of data below.
DISTANCE FROM
GOAL LINE
(IN FEET)
7
12
14
19
21
24
THE
HEIGHT
(IN FEET)
19
90
101
90
55
0
Team B’s egg flew through the air and landed down the field. The group of students tracking the
path of the egg determined that the equation y = –0.8x2 + 19x – 40 represents the path the egg took
through the air, where x is the distance from the goal line and y is the height of the egg from the
ground. (Both measures are in feet.)
When Team C launched an egg with their catapult,
some of the judges found that the graph to the right
shows the path of the egg.
Which team do you think won the contest?
Why?
© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
Team A
1. Using the data from Team A, determine an equation that describes the path of the egg. Describe
how you found your equation.
2. On the graph below, graph the path of Team A’s egg.
3. What is the maximum height that the egg reached? How far was the egg hurled?
Team B
4. Using the equation from Team B, generate a table of values that shows different locations of the
egg as it flew through the air.
x
y
5. On the graph below, graph the path of Team B’s egg.
6. What is the maximum height that the egg reached? How far was the egg hurled?
Team C
7. Using the data from Team C, generate a table of values that shows different locations of the egg
as it flew through the air.
x
y
8. On the graph below, re-graph the path of Team C’s egg.
9. What is the maximum height that the egg reached? How far was the egg hurled?
10. If it is a height contest, which team
wins? How do you know?
11. If it is a distance contest, which team
wins? How do you know?
12. Find a method of determining a winner
so that the team that did not win in
Question 10 or Question 11 would win
using your method.
© 2009 National Council of Teachers of Mathematics
http://illuminations.nctm.org
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
Task #5: Tell an Egg-celent Story
Write a story to describe the three graphs pictured below.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
Task #6: Making Sense of the Three Forms of Quadratic Functions
Often, the standard form of a quadratic is used in projectile motion. For this situation, the equation
h(t) = -½ gt2 + v0t + h0 gives the height of an object at time t, in seconds, for an object with initial
velocity v0 and initial height h0 . “g” is the gravitational constant and is either 9.8 m/s2 or 32 ft/s2. A
simpler form of the equation looks like one of these:
For meters: h(t) = -4.9t2 + v0t + h0
For feet: h(t) = -16t2 + v0t + h0
1. A piece of paper and a hammer are dropped off the top of your school which is 90 feet high.
They are both dropped from a still position (that is, v0 = 0 for both). If we ignore air resistance,
which object, the paper or hammer, hits the ground first? Provide a mathematical argument that
starts by sketching a picture of the graph and concludes with an analysis of the equation.
2. A potato is fired from a spud-gun at a height of 3m and an initial velocity of 25m/s. Write a function
to describe the height of this potato projectile. What's the maximum height of the potato and at what
time does this occur?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
3. Two competing catapults launch pumpkins. Catapult A launches from a starting height of 10ft and
an initial upward velocity of 45ft/sec. Catapult B launches from a starting height of 25ft and an
initial upward velocity of 40ft/sec. Which pumpkin, A or B, achieves a greater maximum height?
Which pumpkin, A or B, is in the air longer?
Is it possible from this scenario to determine the distance traveled horizontally by each pumpkin?
Explain your choices and justify your answers.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
4. The Angry Birds screen shot shows flight paths
of two different birds. Using a straight edge, construct
a coordinate axis system where the center of the
slingshot is at the origin. Carefully assign several
coordinate points to the two parabolas and write an
equation for each. Show which points you used and
which form of the equation you found most helpful.
Using mathematical analysis and your equations, do the two birds hit at the same spot? Why or
why not?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
5. Three points on a parabola are (-3, 0), (6, 0) and (4, -5). Write an equation for this parabola. Which
form is most helpful and why?
6. The vertex of a parabola is (15, -30) and the y-intercept is (0, 25). Is this enough information to write
the equation? If so, do so; if not, explain.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
Task #7: Linear or Quadratic
Based on our work in this lesson and your work in the linear unit (Unit 4), explain in words the
differences between LINEAR and QUADRATIC functions.
How are the structures of the equations different?
How are they similar? Are there similar techniques/processes in graphing and analyzing them? If
so, what are they?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 3
Task #8: Project Planning – Flight of the Gummy Bears
The class is planning a gummy bear launch contest.
If our goal was to hit a target y feet away, what would we need to know? What measurements would
we need to record? What would our launch plan be?
If our goal was to shoot the projectile the longest horizontal distance, what would we need to know?
What measurements would we need to record? What would our launch plan be? How could we
convince the class, using correct mathematical reasoning, that our projectile did indeed travel the
furthest horizontally?
If our goal was to shoot projectile the highest, what would we need to know? What measurements
would we need to record? What would our launch plan be? How could we convince the class, using
correct mathematical reasoning, that our projectile was indeed the highest?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 4
Quadratic Functions
1. Here are 4 equations of quadratic functions and 4 sketches of the graphs of quadratic functions.
A. y = x 2 – 6 x + 8
1.
a.
!
B. y = ( x – 6)( x + 8)
2.
C. y = ( x – 6) 2 + 8
3.
D. y = -( x + 8)( x – 6)
4.
!
Match the equation to its graph and explain your decision.
Equation A matches Graph ....., because
Equation B matches Graph ....., because
Equation C matches Graph ....., because
Equation D matches Graph ....., because
b.
Write the coordinates of the points: P (....,....)
Q(....,....)
R (....,....)
S (....,....)
2. The graph of a quadratic function has a y intercept at (0,5) and a minimum at (3, −4).
a.
Write the equation of its curve.
b.
Write the coordinates of the root(s) of this quadratic function.
Student Materials
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Quadratics Functions
© 2012 MARS, Shell Center, University of Nottingham
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 4
Task #9: What Do You Know?
Graph each quadratic function by first listing and graphing what you know about the function;
then, find and plot 3 more points and sketch the parabola.
(1) f(x) = (x – 3)2 + 1
(2) g(x) = (x + 5)(x + 1)
(3) For each of the following sets of characteristics, first tell whether you know enough
information to write the equation for the parabola. Then, write equations for the ones with enough
known information. Use any form you choose.
(a) Vertex at (6, -4); passes through the point (5, -1)
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 4
(b) Roots at -2 and 1; has a y-intercept of -3
(c) Passes through the points (2, 2) and (8, 5)
(d) Roots at 0 and 5; passes through the point (1, 4.8)
(e) Vertex at (0, 5); has no x-intercepts
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 4
Task #10: Form and Function
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 5
Task #11: The Same Yet Different
The purpose of the next two questions is to use the FORM to answer questions or to perhaps write the
form to answer questions. You may only use a calculator for basic computational facts.
From Illustrative Mathematics:
1. Suppose h(t) = -5t2 + 10t + 3 is a function giving the height of a diver above the water (in meters), t
seconds after the diver leaves the springboard.
a. How high above the water is the springboard? Explain how you know.
b. When does the diver hit the water? Can you find this value without a graphing calculator?
c. At what time on the diver’s descent toward the water is the diver again at the same height as
the springboard?
d. When does the diver reach the peak of the dive? (You don’t know how to move to vertex form
yet, but the idea that the vertex occurs halfway between the x-intercepts is encouraged as a
method for solving.)
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 5
From Illustrative Mathematics:
2. A ball thrown vertically upward at a speed of v0 ft/sec rises a distance d feet in t seconds,
given by d = 6 + v0t – 16t2 . Write an equation (no need to solve) whose solution is:
a. The time it takes a ball thrown at a speed of 88 ft/sec to rise 20 feet.
b. The speed with which the ball must be thrown to rise 20 feet in 2 seconds.
http://www.illustrativemathematics.org/illustrations/437
3. A company’s profit p (in thousands of dollars) from an item depends on the price of the item.
Three different expressions for the profit at a price of d dollars follow:
p = -2d2 + 24d – 54
p = -2(d – 3)(d – 9)
p = -2(d – 6)2 + 18
a. How could you convince someone that the three expressions are equivalent?
b. Which form is most useful for finding:
i. The break-even prices (that is, the price(s) where the profit is 0)? What are those prices, and how
do you know?
ii. The profit when the price is 0? What is that profit, and what does it tell about the business situation?
iii. The price that will yield maximum profit? What is that price?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 5
4. Coyote was chasing Road Runner. Seeing no easy escape, Road Runner jumped off a cliff
towering above the roaring river below. Molly Mathematician was observing the chase and obtained a
digital picture of this fall. Using her mathematical knowledge, Molly modeled the height in feet of Road
Runner’s fall with these quadratic functions, where t is time in seconds:
f(t) = -16t2 + 32t + 48
g(t) = -16(t + 1)(t – 3)
h(t) = -16(t – 1)2 + 64
a. Why does Molly have three equations?
b. Could you convince others that all three of these rules are mathematically equivalent?
c. Which of the rules would be most helpful in answering each of these questions? Explain.
i. What is the maximum height the Road Runner reaches and when will it occur?
ii. When would the Road Runner splash into the river?
iii. At what height was the Road Runner when he jumped off the cliff?
5. Complete the missing entries in the table. Each row represents the same quadratic function.
Standard Form
f(x)
g(x)
h(x)
k(x)
m(x)
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Factored Form
x – 4x – 32
Vertex Form
(x – 2) – 36
2
2
(x – 3)(x + 6)
3x – 10x – 8
2
(x – 2)2 – 49
-(x + 3)2 + 25
X-Intercept(s)
Y-Intercept
Vertex
Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 5
Lesson 5 Independent Practice: Changing Forms
Complete the missing entries in the table. Each row represents the same quadratic function. Sketch
rough graphs of each function and label the points you know.
Standard Form
Factored Form
2(x – 1) – 2
-(x – 5)2 + 4
2
f(x)
g(x)
h(x)
k(x)
j(x)
m(x)
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Vertex Form
(2x – 3)(x + 4)
(x – 2.5)(x + 2.5)
x2 + 16x + 64
x2 + 13x + 42
X-Intercept(s)
Y-Intercept
Vertex
Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Task #12: Flying Marshmallows
Materials: Marshmallow; timer; plastic spoon (optional); a fairly wide open indoor or outdoor
space in which to launch the marshmallows
1. Choose jobs and record assignments here:
Launch Jobs
Analysis Jobs
Marshmallow Tech
Equation Manager
Timer
Graph Manager
Data Recorder
Accuracy Manager
2. Test launching: The marshmallow tech should lie on their back and toss the marshmallow. If
desired, a plastic spoon can be used as a launching mechanism. The launch point should be
essentially at ground level. Practice a few times until a consistent launch path is achievable.
3. Record data: Now record data as accurately as you can about several flights. Record times
to the nearest tenth of a second.
Flight
number
Time marshmallow
was launched
Height of
marshmallow at
Time marshmallow
landed
launch
Height of
marshmallow at
landing
1
2
3
4
5
4. Analyze your marshmallow's flight: Choose the flight that you believe was most accurately
recorded to answer the questions below.
(a) When graphing the flight of your marshmallow, given the data that was recorded, what variable
goes on the x-axis? ________________ How about the y-axis? __________________ What two
points on the graph do you know? _________________________
(b) Calculate the time at which your marshmallow was at its highest point. (Hint: how can you use the
two points you already know to find the peak of the flight?)
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
(c) Sketch a graph of the flight of your marshmallow. Be as precise as you can and label all the
information you currently know.
(d) The factored form of a projectile motion equation looks like this: f(t) = -16(t – t1)(t – t2), where t is the
time in seconds from launch. The function's output is the height of the marshmallow in feet. What do -16,
t1, and t2 represent?
(e) Write the function of your marshmallow's flight in factored form: _____________________________
(f) Convert this function to standard form.
(g) The vertex form of a quadratic equation is y = a(x – h)2 + k where (h, k) is the vertex. Of the letters a,
h, and k, which do you already know?
(h) Use the information you already know and one other point to convert your equation to vertex form.
(i) How high did your marshmallow fly? _____________________________
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Task #13: Flying Marshmallows Follow-Up
Pick ONE flight path from your data to answer the following questions:
In factored form, the flight of your marshmallow looks like: h(t) = -16(t – t1)(t – t2 )
• What do -16, t, t1 and t2 represent?
• Write the equation for the flight of your marshmallow in factored form.
• Write this equation in standard form.
• From either of these forms, what was the maximum height your marshmallow obtained?
• When was this height obtained? Use this information, along with one of your other points to write the
equation for your marshmallows flight in vertex form: y = a(x – h)2 + k.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Task #14: Completing the Square
Method 1 – Algebra Tiles:
1) How does the algebra model tile representation at right
illustrate the product (x + 4)2?
Let’s look at another function: f(x) = x2 + 8x + 10
x
x
1
1
x
x2
x
x
1
1
1
x
x
1
x
1
1
1
1
2) Try to arrange this set of tiles into a PERFECT SQUARE.
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x2
x
x
x
x
x
1
1
1
1
x
1
1
1
1
x
1
1
1
1
x
1
1
1
1
Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
(a) What problems are you running into?
(b) If you ignored for the time being all your “ones,” how many ones in total would you need to make a
PERFECT SQUARE?
(c) Given the “ones” you already have, what could be done to create a perfect square? If I were to
allow you extra tiles, what would you need? Or would you rather take some away?
(d) We can't just change the number of ones in the function; that creates a new function. What else can
we do to the function to keep the net change at zero and therefore keep the function unchanged?
(e) The vertex form of this quadratic is f(x) = (x + 4) 2 – 6. Explain how the process above could help us
arrive at the vertex form.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Method 2 – Area Model (algebra tiles generalized):
3) The squared expression (x + 4)2 is represented geometrically to the
right. Explain/make sense of this model.
x
4
x
x2
4x
4
4x
16
4) Say you are building PERFECT SQUARES and have the following information. In each case, decide
what it is you need to add on to have a perfect square. Draw area models to illustrate your thought
process.
(a) x2 – 8x
(b) x2 – 10x
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
(c) x2 – 3x
(d) x2 + 14x
(e) x2 + 5x
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
5) Now, say you have g(x) = x2 + 6x – 10 and you would like to write it in vertex form.
(a) First you need a perfect square. Ignore the -10 and figure out what it is you need to complete
your perfect square with x2 + 6x.
(b) g(x) must remain ultimately unchanged. How can you keep balance with what you have added to
the function?
(c) Write x2 + 6x – 10 in vertex form.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Task #15: Practice Vertex Form/Complete the Square
Write the following quadratics in vertex form and give the vertex of the quadratic.
1) f(x) = x2 – 12x + 30
2) g(x) = x2 + 16x + 71
3) h(x) = x2 + 2x – 120
4) k(x) = x2 + 4x
5) s(x) = x2 + 3x – 18
6) t(x) = x2 + 7x – 17
7) q(x) = x2 + 9x + 20.25
8) r(x) = (x + 5)(x + 6)
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Task #15 (Cont'd):
1) What value is required to complete the square?
a) x2 + 20x +
b) x2 – 7x +
c) x2 – 4x +
2) Convert each quadratic function to vertex form AND find the coordinates of the max/min point
on its graph.
a) a(x) = x2 + 12x + 11
b) b(x) = x2 – 4x + 7
c) c(x) = x2 – 18x + 74
d) d(x) = x2 – 2x – 48
e) g(x) = x2 – 2x – 8
f) f(x) = x2 + 12x + 20
3) For each of the functions you may use any method you choose to record the information in the table.
Function
y = x2 – 2x – 8
y = x2 + 2x – 33
y = x2 – 8x + 15
y = x2 + 13x + 15
y = x2 + 19x + 60
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x-intercepts
y-intercept
Max or min?
Vertex
Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 6
Task #16: Extension: Piper and Golden Gate Bridge
(1) Piper, the amazing golden retriever, likes to go exploring. Aiming to keep her home more, Stefanie
has decided to fence in part of her yard. She purchased 500 feet of fencing at Lowe's this weekend
and plans to use the back side of her house as one side of the Piper-pen. Stefanie would like to fence
in the largest possible area for Piper. Find the width and length that give Piper the largest possible yard
to play in.
(2) Use vertex form to prove that of all rectangles with a given perimeter, a square has the greatest
area.
(3) The golden gate bridge spans 4,200 feet between towers. The towers supporting the cables are
500 feet high, and at the lowest point, the cable is 8 feet above the roadway. Suppose the point
where the leftmost tower meets the roadway is (0,0). Write a function in vertex form to model the
support cables on the Golden Gate Bridge, where x is the distance from the left tower and y is the
distance from the roadway.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 7
Task #17: Conic Flyer
Describe in general terms how each parameter (slider) changes the graph:
(a) Purple
(b) Red
(c) Blue
1. Which of these parameters affects the range of each parabolic function? Explain.
2. The equation y = 1(x – 0) 2 + 0 or y = x2 is considered the “parent” function for a vertical parabola.
Find five points on the graph of y = x2 and list them below.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 7
3. Fold a sheet of graph paper into fourths, and draw a pair of y- and x- axes in each. Use your five
values to graph y = x 2 on each quarter of the sheet. First predict how the graph of y = x2 would
change for each of the following and then sketch each on the graph paper (without substituting
any points for the new equations below).
(a) y = (x – 2)2 + 4
(b) y = -(x + 3)2 – 1
(c) y = 3(x + 1)2 – 2
1
(d) y = 3(x + 1)2 – 2
4. Determine the range of each function above.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 8
Task #18: Solving Equations
Using any method you choose, solve the following equations.
1. 3x + 7 = 5
2. 3x2 – 5 = 7
3. x2 + 6x = -8
4. 3x2 + 15x – 6 = 0
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 8
Task #19: Angry Birds
An angry bird’s flight path is given by the equation h(t) = 45 + 25t – 16t 2. As a group, use this
information and construct viable arguments to answer the following questions.
(1) Can pigs at heights of 45, 53, and 65 feet be hit?
(2) For the pig(s) that can be hit, at what time will each be hit?
(3) In the game of Angry Birds, you can’t control the height of the slingshot or the effect of gravity, but
you can change how far back and to what angle you pull on the slingshot to fire the bird. This
manipulation affects the initial velocity of the bird. In our original equation the velocity was 25. If you
did not hit one of the pigs at 45, 53, or 65 feet, manipulate the value of b (to signify changing the
slingshot fire) to find a scenario in which you do hit the pig. Construct a viable argument using your
findings for values of b.
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Bridge to College Mathematics . Unit 6 . Lesson 8
Task #20: Two Squares
Solve the quadratic equation using as many different methods as possible.
x2 = (2x – 9)2
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Bridge to College Mathematics . Unit 6 . Lesson 8
Task #21: Solving Quadratic Equations with Tables and Graphs
1. Some highway patrol officers use the formula d = 0.05s2 + 1.1s to predict (or analyze) stopping
distance, d, for speeds, s. For the following equations, determine the solution and explain what
each says about stopping distance and speed. Distance is measured in meters, time in
seconds, and speed in meters/second.
a) 74 = 0.05s2 + 1.1s
b) 42 = 0.05s 2 + 1.1s
c) d = 0.05(13.5) 2 + 1.1(13.5)
d) d = 0.05(9)2 + 1.1(9)
2. The height of a football, in feet, kicked from the ground at time, t, in seconds, can be estimated by
the equation h(t) = 35t – 16t2.
a. Write and solve an equation to show when the football hits the ground at the end of its flight.
b. Regulation for high school, NCAA and the NFL require the goal post to be 10 feet above the
ground. At what times is the ball 10 feet or higher above the ground? Show your work.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 9
Task #22: Solving Quadratic Equations
Solve the following quadratic equations using the method(s) of your choice. Show all work to
make sure others can follow your approach.
1. 3(x – 4)2 – 2 = 25
2. (6x + 5)(2x – 1) = 0
3. 9x2 + 4.7x – 6 = 0
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 9
Task #23: Evaluating Other Students' Thinking
Three students, Jerome, Chelsey, and Travis, were asked to solve the following quadratic equation:
x2 + 4x – 11 = 10
They have shared their processes of solving this problem below:
Jerome: "Factoring is easy! So, I started this problem by first
moving the 10 to set the equation equal to 0 (because before you
factor it has to be equal to 0). That gave me x2 + 4x – 21 = 0. Then
I looked at factors of 21 that added to 4. I came up with -3 and 7.
So my equation now looks like this: (x – 3)(x + 7) = 0. By the zero
product property, I know in order for the product to be 0, one of the
factors must be 0. So x – 3 = 0 is one answer which gives me x =
3 and x + 7 = 0 is another answer which gives me x = -7. My final
answers are x = -7 and 3."
Travis: "Graphing is awesome. It's so easy—all you
have to do is press a few buttons and it's done. But first
you have to get the equation to one side, so I
subtracted 10 from both sides to get x2 + 4x – 21 = 0.
Then I put this in Y1 of my calculator and pressed
graph. I noticed that this parabola crosses the x-axis
twice and the y-axis once. Then I went to my table and
got confused. I saw 0 three times in different places:
when x = 3 and -7 and when y = -21. So I guess there
are three answers: -21, -7, and 3. But I'm not sure."
Chelsey: "I really need to work on completing the square. I
get some of the ideas but need practice so I tried this problem
by completing the square. To start I got everything to one
side. Then I regrouped my terms and left spaces for the ones I
was going to add in. I figured out that I needed 4 to “complete
my square,” so I added 4 and subtracted 4 to keep balance.
So now I have (x + 2)2 – 21 – 4 = 0. I simplified this to
(x + 2)2 – 25 = 0 . Then I started to solve by adding 25 to both
sides to get (x + 2)2 = 25 . Now take the square root of both
sides to get x + 2 = 5 and x + 2 = -5. Solving both of these
gives me my two answers of x = 3 and -7."
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Bridge to College Mathematics . Unit 6 . Lesson 9
1) What do Jerome’s, Travis’s and Chelsey’s methods have in common?
2) Are all three students correct in their reasoning? If not, provide an explanation in words and
pictures to clear up any confusion.
3) Will all three methods work for all quadratic equations? Why or why not?
4) Using the method of your choice, solve the following quadratic equation: x2 – 26 + 6 = 30.
Indicate whose method you chose and why.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 10
Task #24: Roots and Solutions
1. Draw, if possible, a quadratic function that has:
a. zero roots
b. one root
c. two roots
d. three roots
2. Solve 2(x + 3)2 – 5 = 0 without a calculator.
3. Solve 2x 2 + 12x – 13 = 0 without a calculator. (If you can’t do this, don’t worry. We will get to it in
this lesson, but do give it a try.)
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 10
Task #25: Completing the Square to Quadratic Formula
1. Consider 2x2 + 12x – 13 = 0. We will review how, using the strategy of completing the square, we
can change the form of this equation to solve it.
(a) Fill in the blank and describe in words what took place:
2x2 + 12x – 13 = 2(x2 +
) – 13
(b) Which of the following choices is equivalent to 2(x2 + 6x) – 13? Be careful! Consider what
happens if you distribute the 2. Describe in words what took place and why.
(i) 2(x2 + 6x + 9) – 31
(ii) 2(x2 + 6x + 9) – 4
(iii) 2(x2 + 6x + 9) – 22
(iv) 2(x2 + 6x + 9) + 5
(c) Factor the expression inside the parentheses from your answer to part (b) to complete the
following sentence:
2x2 + 12x – 13 = 2(
__
)2 –
(d) Use your answer to (c) to solve 2x2 + 12x – 13 = 0.
We will now do something similar to develop the important and powerful Quadratic Formula, which
allows us to solve EVERY quadratic equation.
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Bridge to College Mathematics . Unit 6 . Lesson 10
2. Suppose we need to solve the equation ax2 + bx + c = 0 for x.
(a) Fill in the blank:
𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 𝑎𝑎(𝑥𝑥 2 + __________) + 𝑐𝑐
(b) Fill in the blank:
𝑎𝑎 �𝑥𝑥 2 + 𝑎𝑎 𝑥𝑥� + 𝑐𝑐 = 𝑎𝑎 �𝑥𝑥 2 + 𝑎𝑎 𝑥𝑥 + 4𝑎𝑎2 � + ________________________
(c) Factor:
𝑏𝑏
𝑏𝑏
𝑏𝑏
𝑏𝑏2
𝑏𝑏2
𝑥𝑥 2 + 𝑎𝑎 𝑥𝑥 + 4𝑎𝑎2 = ______________________________________
(d) Use your answers to (a), (b), and (c) above to complete the following sentence:
𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 𝑎𝑎(𝑥𝑥 + ___________)2 − _________________
(e) Use your answer to (d) to solve the equation ax2 + bx + c = 0. (Think about what you did
to solve 2(x + 3)2 – 31 = 0.)
3. Explain why the part of the formula b2 – 4ac (called the discriminant) tells you, without graphing,
how many real roots the quadratic equation will have.
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 11
Task #26: Rockets
Rockets were assembled from kits by members of an engineering club and were launched from the
ground at the same time. The height y, in feet, of one rocket after t seconds is given by
y = -16t2 + 150t + 5. The height of the other rocket is given by y = -16t2 + 160t. After how many
seconds are the rockets at the same height? What is this height?
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 11
Task #27: Practice Problems: Non-Linear Systems
1. Graph each system below by hand to find the solution(s). Verify your results with a graphing utility.
2
a)
{ yy == x4x++11
b)
{ yy –= xx =–-16x + 9
2
c)
{ 3x2x ––yy==-20
2
d)
{ yy == -1-2x + 4x – 5
2
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Bridge to College Mathematics . Unit 6 . Lesson 11
2. Find the solution(s) to each of the system of equations.
a)
{ yy == xx + 2
2
y = x2 – 4
y = 6x – 13
b)
{
c)
{ 5xx =– 2yy =+156
2
{
y = x2
y=5
e)
{
y = 4x2
y = -3
f)
{ yy == -xx –+4x6x+–48
d)
2
2
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Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 12
Task #28: Gummy Bear Shoot Off
Using a tongue depressor, rubber band and a gummy bear you will devise a contraption to “fire” your
gummy bear. The object of this project is not in the design of your firing device but rather your
mathematical analysis of the flight of your gummy bear. This competition is similar to the egg launch
we looked at in Lesson 3. In fact, you may wish to reference the brainstorming ideas from that lesson.
You are to fire your gummy bear and collect all necessary data. As a group, you will prepare one report
that must include careful mathematical analysis of your gummy bear including equations, graphs,
tables and descriptions. Write AND answer questions about the flight of your gummy bear.
Your final project will be graded according to the rubric and evaluated for mathematical correctness
and completion.
In your report include reflections on the following questions:
• Synthesize what you have learned in this unit. How did you incorporate those ideas into
your mathematical analysis of your gummy bear?
• How do the different forms of a quadratic reveal different information about the flight of your gummy
bear? In answering questions?
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Bridge to College Mathematics . Unit 6 . Lesson 12
Gummy Bear Project Rubric
4
3
2
Graph
Graph is neatly
drawn, accurate,
completely labeled
(including vertex
and intercepts),
and matches the
table.
Graph is neatly
drawn, but may
be missing
several labels
and may have
one or two errors.
Graph is roughly
drawn, labeled
poorly or not at
all, and may not
match the table.
Table
Contains at least 6
points, including
the vertex and the
intercepts. Clear
and neat.
Graph is neatly
drawn, mostly
accurate, almost
completely
labeled (including
vertex and
intercepts), and
matches the
table.
Contains at least
5 points, including
the vertex and the
intercepts. Clear
and neat.
Contains fewer
than 3 points;
difficult to read.
Equations
Equations for all 3
forms are shown,
including the work
to derive each
form. Work is neat
and easy to follow.
No errors present.
Equations for all 3
forms are shown,
including most of
the work to derive
each form. Work
is neat and easy
to follow. 1-2
errors may be
present.
Description
of flight path
Thorough, clear
description
including setup
procedures,
difficulties
encountered,
number of trials,
and comments on
the vertex and
intercepts. Precise
mathematical
language.
Clear description
including setup
procedures,
difficulties
encountered,
number of trials,
and comments on
the vertex and
intercepts. Clear
mathematical
language.
Written
report
Thoughtful,
thorough
synthesis of the
questions asked.
Student was an
engaged partner,
listening to
suggestions of
others and
working
cooperatively
throughout the
project.
Clear synthesis of
the questions
asked.
Contains at least
3 points,
including the
vertex and the
intercepts. Clear
and neat.
Equations for at
least 2 forms are
shown, including
some of the work
to derive each
form. Work is
neat and easy to
follow. Several
errors may be
present.
Description
includes most
elements: setup
procedures,
difficulties
encountered,
number of trials,
and comments on
the vertex and
intercepts.
Mathematical
language may be
imprecise.
Partial synthesis
of the questions
asked.
Student was an
engaged partner
but had trouble
listening to others
and/or working
cooperatively.
Student
cooperated with
others, but
needed
prompting to stay
on-task.
Student did not
work effectively
with others.
Collaboration
1
Only 1 equation is
shown, work is
missing, and/or
major errors are
present.
Description
difficult to
understand
and/or missing
major sections.
Little synthesis of
the questions
asked.
Totals:
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Self
Score
Teacher
Score
Student Manual
Bridge to College Mathematics . Unit 6 . Lesson 12
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