MATHEMATICS 20-2
Quadratic Functions and Equations
High School collaborative venture with
Harry Ainlay, McNally, M. E. LaZerte, Ross Sheppard,
Scona, and W.P. Wagner
Harry Ainlay: Colin Veldkamp
Harry Ainlay: Debby Sumantry
Harry Ainlay: Mathias Stewart
Harry Ainlay: Meriel Hughes
McNally: Enchantra Gramlich
M. E. LaZerte: Monique Merchant
Ross Sheppard: Jeremy Klassen
Ross Sheppard: Tim Gartke
Scona: Joe Johnston
W. P. Wagner: Kiki Brisebois
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 – 2011
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Tasks (on a separate page which can be photocopied & handed out to
students)
Bridge Design for the Masses
Teacher Notes for Transfer Task
Transfer Task
Rubric
Notes and Cautions
Possible Solution
Bridge Design for the Masses
Transfer Task
Rubric
Possible Solution
7
8
15
17
18
28
30
31
STAGE 3 LEARNING PLANS
Lesson #1
Getting to Know You (Characteristics Graphically)
36
Lesson #2
Split Personality (Characteristics Algebraically)
41
Lesson #3
Lesson 3: Solve by Factoring (Solve Equations)
46
Appendix – Worksheets/Keys
Mathematics 20-2
52
Inductive and Deductive Reasoning
Page 3 of 96
Mathematics 20-2
Quadratic Functions and Equations
STAGE 1
Desired Results
Big Idea:
Important real world phenomena can be modeled by quadratic equations and
functions. Understanding the properties of quadratic functions and equations
enhances comprehension of these phenomena. There is an intricate connection
among algebraic functions, the real world and graphs.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …





A quadratic function has a parabolic shape.
Quadratic equations can be solved in a variety of ways.
Quadratic functions and equations can model real world phenomena.
The key features of a quadratic function have contextual significance.
There are relationships among the roots of an equation, the zeros of the
corresponding function and the x-intercepts of the graph of the function.
Essential Questions:





What property of a quadratic function makes it symmetrical? Why does it rise
and fall or fall and rise?
To what extent can a quadratic function describe increasing profit in
business?
What factors affect the path of a projectile?
What causes natural phenomena to be quadratic?
Do we invent math to describe our world or does math describe what is
inherently there?
Mathematics 20-2
Inductive and Deductive Reasoning
Page 4 of 96
Knowledge:
Enduring
Understanding
Specific
Outcomes
Description of
Knowledge
Students will know …
Students will
understand…





A quadratic function
has a parabolic shape.
Quadratic equations
can be solved in a
variety of ways.
Quadratic functions
and equations can
model real world
phenomena.
The key features of a
quadratic function,
expressed in a variety
of ways, have
contextual significance.
There are relationships
among the roots of an
equation, the zeros of
the corresponding
function and the xintercepts of the graph
of the function.
8888
I*RF =
*RF1

*RF2

the general shape of a
parabola
 the vertex, intercepts, domain
& range and axis of symmetry
 the maximum and minimum
values
 how to sketch/plot graphs
 how to identify the vertex of a
quadratic from the standard
form or graph
 identify the opening direction
Students will know …





when to find the x-intercept or
vertex
x-intercepts are zeros and
roots
what solve means
the difference between
functions and equations
how to use the quadratic
formula
when to solve an equation
Relations and Functions
Mathematics 20-2
Inductive and Deductive Reasoning
Page 5 of 96
Skills:
Enduring
Understanding
Specific
Outcomes
Description of
Skills
Students will be able to…
Students will
understand…





A quadratic function
has a parabolic shape.
Quadratic equations
can be solved in a
variety of ways.
Quadratic functions
and equations can
model real world
phenomena.
The key features of a
quadratic function,
expressed in a variety
of ways, have
contextual significance.
There are relationships
among the roots of an
equation, the zeros of
the corresponding
function and the xintercepts of the graph
of the function.
*RF1






convert an equation from standard form to
vertex form
determine the vertex:
o from a graph
o from an equation in standard form
o using technology
determine intercepts:
o from a graph
o from an equation in standard form
o using technology
determine the domain & range from a graph
sketch the graph of a quadratic function with
or without technology
determine the equation of the axis of
symmetry
Students will be able to…
*RF2




solve a quadratic equation by factoring and/or
the quadratic formula
write the function in factored form given the
zeros of a quadratic
determine the number of x-intercepts of a
quadratic function
solve contextual problems
*RF = Relations and Functions
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 6 of 96
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Teacher Notes
There are 2 transfer tasks to evaluate student understanding of the concepts
relating to quadratic functions and equations. A photocopy-ready version is included
for both tasks.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
When work is judged to be limited or insufficient, the teacher makes decisions
about appropriate intervention to help the student improve.
Each student will:

Explain the concepts relating to quadratic functions and equations.
Glossary
factored form – A quadratic function written in the form y = a(x - r)(x - s) , where a,
r, and s are real numbers and a ≠ 0
parabola – A U-shaped curve obtained by graphing a quadratic function
standard form – A quadratic function written in the form y = ax2 + bx + c, where a,
b, and c are real numbers and a ≠ 0
vertex form – A quadratic function written in the form y = a (x – p)2 + q, where a, p,
and q are real numbers and a ≠ 0
vertex of a parabola – The point of intersection of a parabola and its axis of
symmetry
quadratic formula – The formula for determining the roots of a quadratic equation
from its coefficients
Mathematics 20-2
Inductive and Deductive Reasoning
Page 7 of 96
Bridge Design for the Masses - Student Assessment Task
Role: You are working with a team of bridge designers and you have been
asked to create a bridge to cross the North Saskatchewan.
Audience: City Council is eagerly waiting to see what you come up with.
Format: Create a short report to summarize your findings and those of your
colleague.
Topic: Quadratic Functions and Equations
Scenario A
You have been asked to design a bridge to span the North Saskatchewan,
and you are leaning towards a parabolic shape, but you have some research
to do first.
The bridge must span 120 m to get from one side of the river to the other, in
the spot requested by City Council.
Your first task is to determine the equation of your quadratic in the form
, where r and s are the x-intercepts of the graph.
You will need to identify some properties of your model. Make sure that in
your report you:
 Identify the axis of symmetry.
 Find the y-coordinate of the vertex and determine whether it is a
maximum or a minimum.
 Identify the domain and range
 Include a sketch of your model on the grid provided
 Move the axes to create at least two different sets of r and s values.
Determine the new equation and verify that the maximum or minimum
value is the same as that in the first equation.
 Comment on the feasibility of your model and what, if anything could
be changed to improve it
Bridge Design for the Masses (continued)
Scenario B
A colleague is also working on a bridge model. His quadratic model is
y = a(x 2 - 25x -12) . He is having difficulty with this model, but he believes
the problem can be fixed by altering the value of a. Your colleague has asked
for your help.
Choose two values for a in the interval –5 to –1. For each of these new
equations find the x-intercepts of the graph. In one case use the calculator to
find these values, and in the other use the quadratic formula. Round your
answers to the nearest hundredth in both cases. What do you notice about
your x-intercepts in each case, and what does this mean?
For both of these models find the maximum height of the structure. Comment
on the impact of a in the quadratic function.
Your colleague also needs to determine an equation that will restrict the
maximum height range to 25 m – 30 m. Generate a quadratic equation that
meets this condition, while maintaining the same x-intercepts.
Bridge Design for the Masses (continued)
Scenario C
Your bridge inspector colleague received plans for the construction of a walkway bridge
to be known as the PiederManz Bridge, and has given you a copy of his notes. There
are two parts to the design; a bridge support which has a parabolic shape, and a linear
walkway intersecting the support. He has concerns that the walkway will be unsafe.
The following artist’s rendering shows the walkway bridge over a small market between
a river and road.
B
A
Source: http://www.newcastle-guide.co.uk/images/stories/tyne-bridges042.JPG
Bridge Inspector Notes:
The maximum height of the bridge support is to be 55 m above the ground, and the
distance between the supports on the ground is to be about 160 m.
The equation of the support is given by h(x) = –111280(x – 80)2 + 55, where h is the
height above ground, in meters, and x is the distance, in meters, from the bottom of the
left end of the support.
The walkway intersects the bridge support at a height of 32 m above the ground (point
A at the left), and intersects it at a height of 37 m above ground at the right (point B).
I’m not sure the distance between A and B is short enough! I’m also not sure how steep
this walkway is, and I’m fearful that some elderly folks may slip and fall in the winter.
You know that the length of the walkway between the supports needs to be less than
100 m, and that the grade of the walkway needs to be less than 5%. Based on your
colleague’s notes, comment on the safety of the bridge design.
Bridge Design for the Masses (continued)
Scenario A: Show your work here.
y = – (x – r)(x – s)
Bridge Design for the Masses (continued)
A
B
source: http://t3.gstatic.com/images?q=tbn:ANd9GcQ0OBEsEh86rnP4b2ojvnzEHPVz7hQegYTNAtN3A4HpGt7ug77Lg
Discuss your model’s feasibility and your suggestions here:
Bridge Design for the Masses (continued)
Scenario B: Show your work here.
y = a(x2 – 25x – 12)
Comment on the impact of a.
State the equation you found:
Bridge Design for the Masses (continued)
Scenario C: Show your work here.
h(x) = –111280(x – 80)2 + 55
Comment on your colleague’s design.
Bridge Design for the Masses (continued)
Glossary
factored form – A quadratic function written in the form y = – (x – r)(x – s), where a, r,
and s are real numbers and a ≠ 0
parabola – A U-shaped curve obtained by graphing a quadratic function
standard form – A quadratic function written in the form y = ax2 + bx + c, where a, b,
and c are real numbers and a ≠ 0
vertex form – A quadratic function written in the form y = a (x – p)2 + q, where a, p,
and q are real numbers and a ≠ 0
vertex of a parabola – The point of intersection of a parabola and its axis of
symmetry
quadratic formula – The formula for determining the roots of a quadratic equation
from its coefficients
Assessment
Bridge Design for the Masses Rubric
Component
Description of Requirements
- Scenario A
- Appropriate factored form of equation
- Identifies axis of symmetry, vertex, domain and
range
Identifies a valid second equation with verification
Mathematical
Content
- Scenario B
- Correct x-intercepts found in prescribed manner
- Appropriate equation (a-value) stated
- Scenario C
- Determines proper length and grade of walkway
Presentation
- Report is easy to read
of Data
- Math work is complete and easy to follow
- Graph is complete and accurate (scale, labels,
neatly drawn)
Explanations
- Comment/recommendations on the feasibility of
of Choices
the model of the equation in Scenario A.
- Conclusions about x-intercepts and impact of a in
Scenario B are appropriate.
- Conclusions about walkway are appropriate in
Scenario C
Assessment
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
Bridge Design for the Masses (continued)
Level
Criteria
Math
Content
Scenario
A
Excellent
4
All required
elements are
present and
correct
Math
Content
Scenario
B
All required
elements are
present and
correct
Math
Content
Scenario
C
All required
elements are
present and
correct
Presents
Data
Presentation
of data is
clear,
precise and
accurate
Explains
Choices
Provides
insightful
explanations
Proficient
3
All required
elements are
present but
may contain
minor errors
Adequate
2
Some
required
elements are
missing, or
contain
major errors
All required
Some
elements are required
present but
elements are
may contain missing, or
minor errors contain
major errors
All required
Some
elements are required
present but
elements are
may contain missing, or
minor errors contain
major errors
Presentation Presentation
of data is
of data is
complete
simplistic
and
and
unambiguo plausible
us
Provides
Provides
logical
explanations
explanations that are
complete
but vague
Limited
1
Most
required
elements are
missing or
incorrect
Insufficient
Blank
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Most
required
elements are
missing or
incorrect
No score is
awarded as
there is no
evidence
given
Presentation
of data is
vague and
inaccurate
Presentation
of data is
incomprehe
nsible
Provides
explanations
that are
incomplete
or
confusing.
No
explanation
is provided
When work is judged to be limited or insufficient, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 20-2
Inductive and Deductive Reasoning Page 16 of 96
Bridge Design for the Masses
Notes and Cautions
The thought in designing this transfer task was that students would complete two of
the three scenarios.
In Scenario A students may need some direction to adjust the (vertical) scale of the
graph to accommodate the height of the vertex.
In Scenario B encourage students to use logic and verification, and to round their avalues to the nearest hundredth.
In Scenario C this section can be assigned as is, or can be adjusted. One suggestion
is to add a recommendation to the colleague (so that the grade problem is fixed).
If a student attempts to find the x-value of the walkway algebraically, they may need
some guidance as each y-value produces an extraneous root.
First Possible Solution to Bridge Design for the Masses
Scenario A
Mathematics 20-2
Quadratic Functions and Equations
Page 18 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 19 of 96
Scenario B
Mathematics 20-2
Quadratic Functions and Equations
Page 20 of 96
Scenario C
Mathematics 20-2
Quadratic Functions and Equations
Page 21 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 22 of 96
Second Possible Solution to Bridge Design for the Masses
Scenario A
Mathematics 20-2
Quadratic Functions and Equations
Page 23 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 24 of 96
Scenario B
Mathematics 20-2
Quadratic Functions and Equations
Page 25 of 96
Scenario C
Mathematics 20-2
Quadratic Functions and Equations
Page 26 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 27 of 96
Public Awareness Campaign - Student Assessment Task
Situation
There has been an anti quadratics movement afoot in the school system
organized by a strongly opinionated parent group. The parent group wants to
remove quadratic equations and functions from the Alberta Mathematics
Curriculum for the following reasons:
 There are limited applications of quadratic functions.
 Quadratic equations are too difficult to solve.
 The information gathered from a quadratic function is not relevant to the real
world.
 People function in the real world without graphs of quadratics.
Goal
Create a public awareness campaign to stress the importance of quadratics in
mathematics and real world applications.
Mathematics 20-2
Quadratic Functions and Equations
Page 28 of 96
Public Awareness Campaign (continued)
Role:
You are to choose one of the following methods to support your public relations
campaign.
Oral Projects
Products
Multimedia
Newscast - Presentation
and Planning
Making A Brochure
PowerPoint Presentation
Making A Poster
Video Presentation
Puppet Show
Story Telling
Newspaper (ad, article or
editorial)
Video - Talk Show
Audience
You are to present to the mock parent group and convince them that quadratic functions
and equations are a valuable asset to the Alberta Mathematics Curriculum.
Product
Your product should include:
Four examples of parabolas in the real world. Include visuals.
For one of your examples:




Identify the vertex and its significance.
Identify the y-intercept and its significance.
Identify the number of x-intercepts and their significance.
Determine the domain and range within the context of your example.
Two quadratic equations in standard form:
 Calculate the x-intercepts using a minimum of two methods. Clearly show
(or explain) all steps.
Generate graphs of two quadratic functions, one in standard form and one in vertex form,
using two of the following methods.
 table of values
 graphing calculator
 plot x-intercepts, y-intercept and vertex
Public Awareness Campaign Rubric
Level
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Criteria
Performs
Performs precise and
Performs focused and Performs
Performs
Calculations
explicit calculations.
accurate calculations. appropriate and
superficial
generally accurate and irrelevant
States two
 Correctly identified
 Correctly identified
quadratic
calculations.
calculations.
four intercepts
two or three
equations in
 Correctly
 Some
intercepts
standard form:
identified one
correct
Calculates the
intercept
algebra but
x-intercepts
no
using a
correctly
minimum of
identified
two methods.
intercepts
Clearly
explains all
steps.
Indentifies
 Identifies the vertex  Was able to correctly  Missed two
 Missed
and explains
and its significance
identify three of the
important key
three
significance
including identifying
key points
points
important
of key points
as a maximum or
key points
of quadratics.
minimum
 Identifies the yintercept and its
significance
 Identifies the number
of x-intercepts and
their significance
 Determines the
domain and range
within the context of
the example
Communicate Develops a
Develops a
Develops a
Develops an
s findings,
compelling and
convincing and
predictable
unclear
include
precise presentation
logical presentation
presentation that
presentation
visuals and
that fully considers
that mostly considers partially considers with little
graphs
consideration
the purpose and
the purpose and
the purpose and
audience; uses
audience; uses
audience; uses
of purpose and
appropriate
appropriate
some appropriate
audience;
mathematical
mathematical
mathematical
uses
inappropriate
vocabulary, notation
vocabulary, notation
vocabulary,
and symbolism
and symbolism
notation and
mathematical
symbolism
vocabulary,
notation and
symbolism
Real life or
Includes four real life
Includes three real life Includes real life
Includes real
real work
applications clearly
applications clearly
applications not
life
application
linked to the real
clearly linked to the applications
linked to the real
are included
world or four
with limited
world.
real world
applications not
applications to
clearly linked to the
the real world
real world
Mathematics 20-2
Quadratic Functions and Equations
Insufficient/
Blank*
No score is
awarded
because there
is no evidence
of student
performance
No data is
presented
No findings
are
communicated
No real life
examples are
included
Page 30 of 96
Possible Solutions to Public Awareness Campaign
Screenshot of Brochure Exemplar
SmartBoard Exemplar (MS Word)
SmartBoard Exemplar (PDF)
 files were added to the EPSB Understanding by Design share site
Screenshot of SmartBoard Exemplar
Public Awareness Campaign Exemplar (Notebook)
 file was added to the EPSB Understanding by Design share site
Mathematics 20-2
Quadratic Functions and Equations
Page 31 of 96
Screenshot of Video Exemplar (Presentation of SmartBoard file)
Video Exemplar
(small file): http://dl.dropbox.com/u/6062352/Quadratics Transfer Task Exemplar. Smaller.mp4
(large file): http://dl.dropbox.com/u/6062352/Quadratics Transfer Task Exemplar.m4v
Public Awareness Video Script
 file was added to the EPSB Understanding by Design share site
Mathematics 20-2
Quadratic Functions and Equations
Page 32 of 96
STAGE 3
Learning Plans
Lesson 1
Characteristics: Graphically
STAGE 1
BIG IDEA: Important real world phenomena can be modeled by quadratic equations and functions. Understanding the
properties of quadratic functions and equations enhances comprehension of these phenomena. There is an intricate
connection among algebraic functions, the real world and graphs.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …







A quadratic function has a parabolic shape.
Quadratic functions and equations can model real
world phenomena.
The key features of a quadratic function, expressed in
a variety of ways, have contextual significance.
There are relationships among the roots of an
equation, the zeros of the corresponding function and
the x-intercepts of the graph of the function.
What property of a quadratic function makes it
symmetrical? Why does it rise and fall or fall and rise?
What factors affect the path of a projectile?
Do we invent math to describe our world or does math
describe what is inherently there?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …










the general shape of a parabola
the vertex, intercepts, domain & range and axis of
symmetry
the maximum and minimum values
how to sketch/plot graphs
how to identify the vertex of a quadratic from the
standard form or graph
identify the opening direction
when to find the x-intercept or vertex
x-intercepts are zeros and roots
the difference between functions and equations



determine the vertex:
o from a graph
o from an equation in standard form
o using technology
determine intercepts:
o from a graph.
o from an equation in standard form
o using technology
determine the domain & range from a graph
sketch the graph of a quadratic function with or
without technology
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 20-2
Quadratic Functions and Equations
Page 33 of 96
Lesson Summary




Introduce quadratic functions.
Given a graph of a quadratic function, find the y-intercept, vertex, x-intercepts, domain
and range.
Given a quadratic function, make a table of values and graph. Look at symmetry of a
parabola and introduce axis of symmetry.
Given a quadratic function, use a calculator to determine the vertex, x-intercepts, yintercept, axis of symmetry, and domain and range.
Lesson Plan
Hook
Show the waterslide video. Explain this is a real life example of a nonlinear relationship. This
unit is focusing on a nonlinear function called a quadratic.
Waterslide video
Source: http://www.youtube.com/watch?v=Fyo-nhoNW54
Mathematics 20-2
Quadratic Functions and Equations
Page 34 of 96
Lesson Goal
Students will be able to identify the vertex, axis of symmetry, intercepts, domain and range of a
quadratic function given a graph or a function (standard and vertex form).
Activate Prior Knowledge
Display a graph of a quadratic function that opens down (no scale on grid) and ask students to
come up with a scenario that could be represented by the graph. Introduce the term parabola
and emphasize that this shape is parabolic.
Lesson
Part 1
Give a picture of Angry Bird.
Source: http://media.photobucket.com/image/angry%20birds%20projectile/leonography/News/angrybirdT02.jpg
What do you want to know about this bird? Potential questions:
 How high did the bird go?
 How high was it when it was launched?
 Where will it land? How long was it in the air?
Mathematics 20-2
Quadratic Functions and Equations
Page 35 of 96
Give a picture of Angry Bird with a grid superimposed on it and a graph of the path (integral
key points). Can we answer these questions using this graph?
or
Source: John Scammell’s iPad
What are the key points on this curve?
Introduce the term vertex and discuss vertex, domain, range and intercepts.
Mathematics 20-2
Quadratic Functions and Equations
Page 36 of 96
Part 2
Show second video clip with different contexts for parabolas.
source: http://qwickstep.com/search/skateboarding-half-pipe.html
Show the picture and discuss the structure of the half pipe and/or the path of the skateboarder
from one side to the other.
The function that approaches this situation is h = 4(t – 1.5)2 + 3 and is shown in the graph
below.
1.
2.
3.
4.
5.
6.
What do h and t represent in the function?
How does that relate to the graph?
How high is he in the picture?
What’s the lowest height on the graph?
How long does it take him to reach the other side?
What are the domain and range for this function?
DI Suggestion
Discuss why there are no x-intercepts for
this situation.
Discuss why we have to say that a halfpipe approaches a quadratic function.
Mathematics 20-2
Quadratic Functions and Equations
Page 37 of 96
Give students a function in standard form.
Example: y = – 2(x – 5)2 + 50
Provide x-values for the table of values that will provide students with symmetry cues. Have
students determine the y-values, then sketch the graph using the table of values.
Talk about the symmetry shown in the table of values. What are the key points for this curve?
Introduce the axis of symmetry.
Give students a graph (on labelled grid) that opens up with no x-intercepts. Ask them to
determine the key points (x-intercepts, y-intercept, vertex). We’ve looked at a graph with 2 xintercepts and one with no x-intercepts, are there any other possibilities?
Draw a sketch.
DI Suggestion
Consider a discussion about graphs and
the quadrants containing data points.
Part 3
Show a video clip of a soccer goal. A shortened version of this video was added to the same
folder as this file.
Before showing the function consider
discussing the characteristics of the graph
of the function.
If the ball is kicked at (0, 0) and lands at
(4, 0) answer the following
 What is the axis of symmetry?
 Where is the vertex?
 Is the y-coordinate of the vertex a
maximum or minimum?
(RF 1.3)
L1 Soccer Goal
 file was added to the EPSB Understanding by Design share site
 original source: http://www.youtube.com/watch?v=X72R9yu_4hU&feature=fvw
The path of the ball is represented by the function y = –2x2 + 8x, where y represents the height
of the ball off the ground in metres, and x represents time in seconds. Instruct how to find the
key points using the calculator. Look at questions like how high is the ball at 1.6 s and/or at
what time does the ball reach a height of 5.5 m? Discuss applicable domain and range.
DI Suggestion
Consider making a mini book with
calculator functions.
Mathematics 20-2
Quadratic Functions and Equations
Page 38 of 96
Going Beyond
Resources
Math 20-2 (Nelson: sec 6.1, 6.2, 6.4, 6.5 page(s) 322-336, 354-382)
Watch a video of different parabolic scenarios.
Source: http://www.learnalberta.ca/content/mejhm/html/video_interactives/polynomials/movie.html
Supporting
Assessment
Mathematics 20-2
Quadratic Functions and Equations
Page 39 of 96
Glossary
axis of symmetry - A line that separates a figure into two equal parts
domain - All possible independent values for a relation
maximum value - The greatest value of a dependent variable of a relation
minimum value - The least value of a dependent variable of a relation
parabola – A U-shaped curve obtained by graphing a quadratic function
quadratic equation – An equation of the form ax2 + b x + c = 0, where a, b, and c are real
numbers and a ≠ 0
quadratic function – A function that may be written in the form y = ax2 + b x + c, where a, b,
and c are real numbers and a ≠ 0
quadratic relation – A relation that can be written in the standard form ax2 + b x + c = 0,
where a, b, and c are real numbers and a ≠ 0 [Math 20-2 (Nelson, page 517)]
range - All possible dependent values for a relation
root – A solution of an equation
vertex of a parabola – The point of intersection of a parabola and its axis of symmetry
x-intercept - The horizontal value of the point where the curve intersects the horizontal axis
y-intercept - The vertical value of the point where the curve intersects the vertical axis.
zero – The value of the independent variable of a function for which the function equals zero
Other
Mathematics 20-2
Quadratic Functions and Equations
Page 40 of 96
Lesson 2
Characteristics: Algebraically
STAGE 1
BIG IDEA: Important real world phenomena can be modeled by quadratic equations and functions.
Understanding the properties of quadratic functions and equations enhances comprehension of these
phenomena. There is an intricate connection among algebraic functions, the real world and graphs.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …

 A quadratic function has a parabolic shape.
 Quadratic functions and equations can model
real world phenomena.
 The key features of a quadratic function,
expressed in a variety of ways, have contextual
significance.
 There are relationships among the roots of an
equation, the zeros of the corresponding
function and the x-intercepts of the graph of the
function.



What property of a quadratic function makes it
symmetrical? Why does it rise and fall or fall
and rise?
What factors affect the path of a projectile?
What causes natural phenomena to be
quadratic?
Do we invent math to describe our world or
does math describe what is inherently there?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …










the general shape of a parabola
the vertex, intercepts, domain & range and
axis of symmetry
the maximum and minimum values
how to sketch/plot graphs
how to identify the vertex of a quadratic from
the standard form or graph
identify the opening direction
when to find the x-intercept or vertex
x-intercepts are zeros and roots
when to solve an equation








Mathematics 20-2
determine the vertex:
o from a graph
o from an equation in standard form
o using technology
determine intercepts:
o from a graph.
o from an equation in standard form
o using technology
determine the domain & range from a graph
sketch the graph of a quadratic function with
or without technology
determine the equation of the axis of
symmetry.
solve a quadratic equation by factoring and/or
the quadratic formula
write the function in factored form given the
zeros of a quadratic
determine the number of x-intercepts of a
quadratic function
solve contextual problems
Quadratic Functions and Equations
Page 41 of 96
Lesson Summary

In this lesson students will explore the effects of the parameters of quadratic functions in
vertex form, standard form and factored form.
Lesson Plan
Lesson Goal
 Students will see how the different forms of equations relate to the characteristics of
quadratic functions.
 Students will discover the effects each parameter (for each form) has on the graph.
Activate Prior Knowledge
What do you see activity
 Using the image of a quadratic function below, ask the students “What do you see?”
 A Think-Pair-Share is a possible way to promote student discussion about the graph.
Student discussion will hopefully lead to a discussion about the characteristics of a
quadratic function (vertex, intercepts, axis of symmetry, domain and range.)
Mathematics 20-2
Quadratic Functions and Equations
Page 42 of 96
Lesson
Students will investigate the properties for vertex, standard, and factored form of the quadratic
function
Goal is to see that:
 Vertex form is useful for determining the vertex, direction of opening, and shape of the
graph.
 Standard form is useful for determining the y-intercept, direction of opening, and shape
of the graph.
 Factored form is useful for determining the x-intercepts, direction of opening, and shape
of the graph.
At the end of the each section you may use an applet to visually reinforce the students
learning.
1. Vertex Form: y = a(x – p) 2 + q
source: http://members.shaw.ca/jreed/math20-2/ubd/quadratic.htm
L2 Investigate Vertex Form and Key


copy was added to Appendix
file was added to the EPSB Understanding by Design share site
2. Jig saw vertex, standard, and factored forms activity:
Students will practice graphing quadratic functions from the different forms.
The goal of the Jig Saw Activity is for students:
Mathematics 20-2
Quadratic Functions and Equations
Page 43 of 96


to see that the different forms of a quadratic function can all result in the same
graph
to understand the advantages and disadvantages of each form
L2 Jig Saw Equation Forms and Key


copy was added to Appendix
file was added to the EPSB Understanding by Design share site
3. Jig saw applications activity:
Students will practice solving different types of application questions involving
quadratics.
L2 Jig Saw Applications and Key


copy was added to Appendix
file was were added to the EPSB Understanding by Design share site
Instructions
Arrange students in groups of 4.
Each student will receive a different question (Circle, Triangle, Square, Star) and
individually work on their problem.
Similar shapes will meet, compare their work, and become experts on their problem.
Bring students back to their original groups and each student will present their problem
and solution to the entire group.
4. Graphs, equations, and characteristics matching activity
Students complete the activity to show their understanding of the characteristics of
quadratic functions and the different forms of the equations of quadratic functions
L2 Matching Activity with Key
 copy was added to Appendix
 file was added to the EPSB Understanding by Design share site
5. Finding equations of quadratic functions
Now that students are familiar with the different forms of quadratic functions they can
practice finding the equations.
Consider using the examples from the first lesson to practice finding equations.
Going Beyond
Resources
Math 20-2 (Nelson: sec 6.1 – 6.5, page(s) 322-382)
Mathematics 20-2
Quadratic Functions and Equations
Page 44 of 96
Supporting
Assessment
Glossary
axis of symmetry - A line that separates a figure into two equal parts
domain - All possible independent values for a relation
factored form – A quadratic function written in the form y = a(x – r) (x – s), where a, r, and s
are real numbers and a ≠ 0
maximum value - The greatest value of a dependent variable of a relation
minimum value - The least value of a dependent variable of a relation
parabola – A U-shaped curve obtained by graphing a quadratic function
quadratic formula – The formula for determining the roots of a quadratic equation from its
coefficients
range - All possible dependent values for a relation
standard form – A quadratic function written in the form y = ax2 + bx + c, where a, b, and c are
real numbers and a ≠ 0
vertex form – A quadratic function written in the form y = a (x - p)2 + q, where a, p, and q are
real numbers and a ≠ 0
vertex of a parabola – The point of intersection of a parabola and its axis of symmetry
x-intercept - The horizontal value of the point where the curve intersects the horizontal axis
y-intercept - The vertical value of the point where the curve intersects the vertical axis.
zero – The value of the independent variable of a function for which the function equals zero
Other
Mathematics 20-2
Quadratic Functions and Equations
Page 45 of 96
Lesson 3
Lesson 3: Solve by Factoring (Solve Equations)
STAGE 1
BIG IDEA: Important real world phenomena can be modeled by quadratic equations and functions.
Understanding the properties of quadratic functions and equations enhances comprehension of these
phenomena. There is an intricate connection among algebraic functions, the real world and graphs.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …






A quadratic function has a parabolic shape.
Quadratic equations can be solved in a
variety of ways.
Quadratic functions and equations can model
real world phenomena.
The key features of a quadratic function,
expressed in a variety of ways, have
contextual significance.
There are relationships among the roots of an
equation, the zeros of the corresponding
function and the x-intercepts of the graph of
the function.




What property of a quadratic function makes it
symmetrical? Why does it rise and fall or fall
and rise?
To what extent can a quadratic function
describe increasing profit in business?
What factors affect the path of a projectile?
What causes natural phenomena to be
quadratic?
Do we invent math to describe our world or
does math describe what is inherently there?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …









axis of symmetry
how to sketch/plot graphs
identify the opening direction
when to find the x-intercept or vertex
x-intercepts are zeros and roots
what solve means
the quadratic formula
when to solve an equation






Mathematics 20-2
determine the vertex:
o from a graph
o from an equation in standard form
o using technology
determine intercepts:
o from a graph.
o from an equation in standard form
o using technology
sketch the graph of a quadratic function with
or without technology
solve a quadratic equation by factoring and/or
the quadratic formula
write the function in factored form given the
zeros of a quadratic
determine the number of x-intercepts of a
quadratic function
solve contextual problems
Quadratic Functions and Equations
Page 46 of 96
Lesson Summary
 Zeros are the x-intercepts of the graph of the corresponding quadratic function and the
roots of the corresponding quadratic equation.
 Methods for finding the roots of (solving) a quadratic equation include:
 graphing
 factoring
 quadratic formula
 rectangles and algebra tiles
 Checking/verifying the root(s) require(s) substitution or the zero feature on the graphing
calculator.
Lesson Plan
Hook
(x - 3)(x - 5) = 0
What do we call this form of a quadratic equation?
Lesson Goal
Students should be able to determine, by factoring, the roots of a quadratic equation and verify
by substitution.
Activate Prior Knowledge
Review rectangles and algebra tiles. The following applet shows 3 polynomials with a leading
coefficient of 1. Two buttons give opportunity to explore factoring with and without tiles for
leading coefficients of 1 (Natural/Integers). The last button provides opportunity to explore
expressions where the leading coefficient is not 1.
source: http://staff.argyll.epsb.ca/jreed/math9/strand2/factor1.htm
It would be useful for students to be able to switch between polynomial, rectangular
representation, grid and factors. An assignment with 4 columns with one example in each
column should help students show multiple representations of the same expression.
Mathematics 20-2
Quadratic Functions and Equations
Page 47 of 96
L3 Rectangles and Algebra Tiles Completion Assignment
 copy was added to Appendix
 file was added to the EPSB Understanding by Design share site
Factoring review examples:
Factoring Trinomials
Example: 6x2 + 14x - 12
2(3x2 + 7x – 6)
GCF First!
Factor Pictorially
Factor Symbolically
Difference of Squares
Example: 27x2 - 75
2
GCF First!
2(9x – 25)
Factor Pictorially
Factor Symbolically
L3 Factoring Review Examples
 copy was added to Appendix
 file was added to the EPSB Understanding by Design share site
Mathematics 20-2
Quadratic Functions and Equations
Page 48 of 96
Lesson
Discuss with students the
 possible values for a & b in ab = 0
 solutions for 2b = 0 and a(5) = 0
 solutions for 2(x + 3) = 0 and (x + 7)(5) = 0
 solutions for (x - 3)(x - 5) = 0
 solutions for x(x - 5) = 0
Emphasize that roots and solutions are equivalent terms.
Students should enter the function y = (x - 3)(x - 5) into their graphing calculators and
determine the zeros. Remind students that zeros of a function are the x-intercepts of the graph
of the function.
To help students link the terms solutions, roots, zeros and x-intercepts have them look up the
textbook (alternatively, display on screen) definition for zeros: “In a function, the value of the
variable that makes the value of the function equal to zero.” (Nelson)
 Ask “How do y = (x - 3)(x - 5) and (x - 3)(x - 5) = 0 relate to each other?” Differentiate
between the ordered pairs of zeros and x-intercepts with the x-values of the solutions and
roots.
 Explain that solving means making y = 0 or making the function equal to zero
 Emphasize that in order to solve an equation, it must be equal to 0.
Quadratic Formula
It may useful for students to see factoring visually and algebraically. The following applet
shows the same quadratic solved by quadratic formula, graphing and factoring.
source: http://staff.argyll.epsb.ca/jreed/math9/strand2/factor2.htm
L3 Solving Using the Quadratic Formula
 copy was added to Appendix
 file was added to the EPSB Understanding by Design share site
Mathematics 20-2
Quadratic Functions and Equations
Page 49 of 96
Show students how to solve these examples: x2 - 2x - 120 = 0 and -3.4x2 + 10.2x + 61.2.
Discuss how students can check their answers (graphing calculator, verifying algebraically
by substitution, …).
Split the students into groups of 4 to solve a package of equations.
L3 Solving Equations Questions and Key
 copy was added to Appendix
 file was added to the EPSB Understanding by Design share site
Going Beyond
The text resource has a number of contextual problems.
Resources
Math 20-2 (Nelson: sec 6.3, page(s) 405-413)
Math 20-2 (Nelson: sec 7.2, page(s) 337-349)
Supporting
Assessment
Glossary
axis of symmetry - A line that separates a figure into two equal parts
domain - All possible independent values for a relation
factored form – A quadratic function written in the form y = a(x - r)(x - s), where a, r, and s are
real numbers and a ≠ 0
maximum value - The greatest value of a dependent variable of a relation
minimum value - The least value of a dependent variable of a relation
parabola – A U-shaped curve obtained by graphing a quadratic function
Mathematics 20-2
Quadratic Functions and Equations
Page 50 of 96
quadratic equation – An equation of the form ax2 + b x + c = 0, where a, b, and c are real
numbers and a ≠ 0
quadratic formula – The formula for determining the roots of a quadratic equation from its
coefficients
range - All possible dependent values for a relation
root – A solution of an equation
standard form – A quadratic function written in the form y = ax2 + bx + c, where a, b, and c are
real numbers and a ≠ 0
vertex form – A quadratic function written in the form y = a (x - p)2 + q, where a, p, and q are
real numbers and a ≠ 0
vertex of a parabola – The point of intersection of a parabola and its axis of symmetry
x-intercept - The horizontal value of the point where the curve intersects the horizontal axis
y-intercept - The vertical value of the point where the curve intersects the vertical axis.
zero – The value of the independent variable of a function for which the function equals zero
Other
Mathematics 20-2
Quadratic Functions and Equations
Page 51 of 96
Appendix
Copies of worksheets for Lessons 2 – 3 follow:
Lesson 2 Investigate Vertex Form and Key
Lesson 2 Jig Saw Applications and Key
Lesson 2 Jig Saw Equation Forms and Key
Lesson 2 Matching Activity with Key
Lesson 3 Rectangles and Algebra Tiles Completion Assignment
Lesson 3 Factoring Review Examples
Lesson 3 Solving Using the Quadratic Formula
Lesson 3 Solving Equations Questions and Key
Mathematics 20-2
Quadratic Functions and Equations
Page 52 of 96
M20-2 Lesson 2 Investigate Vertex Form
Investigate 𝑦 = (𝑥 − ℎ)2
For the following equations 𝑦 = 𝑥 2
𝑦 = (𝑥 − 2)2
𝑦 = (𝑥 + 1)2 ,
1. Complete the table of values.
2. Plot the points on the grid using different colors for each equation.
3. For each equation, complete the chart below the grid.
𝒚 = (𝒙 − 𝟐)𝟐
x
y
0
1
2
3
4
𝒚 = 𝒙𝟐
x
-2
-1
0
1
2
y
𝒚 = 𝒙𝟐
𝒚 = (𝒙 − 𝟐)𝟐
𝒚 = (𝒙 + 𝟏)𝟐
x
y
-3
-2
-1
0
1
𝒚 = (𝒙 + 𝟏)𝟐
Vertex
Maximum or
minimum value
y-intercept
x-intercept(s)
Equation of the Axis
of Symmetry
Domain
Range
In General: The graph of y  ( x  h) 2 is the graph of y  x 2 is moved along the x-axis…
Left h units if: _____________.
Right h units if: ___________.
Investigate 𝑦 = 𝑥 2 + 𝑘
For the following equations 𝑦 = 𝑥 2
𝑦 = 𝑥2 + 3
𝑦 = 𝑥 2 − 2,
1. Complete the table of values.
2. Plot the points on the grid using different colors for each equation.
3. For each equation, complete the chart below the grid.
Mathematics 20-2
Quadratic Functions and Equations
Page 53 of 96
𝑦 = 𝑥2 + 3
x
y
0
1
2
3
4
𝒚 = 𝒙𝟐
x
-2
-1
0
1
2
y
𝑦 = 𝑥2 − 2
x
y
-3
-2
-1
0
1
𝑦 = 𝑥2 + 3
𝒚 = 𝒙𝟐
𝑦 = 𝑥2 − 2
Vertex
Maximum or
minimum value
y-intercept
x-intercept(s)
Equation of the Axis
of Symmetry
Domain
Range
In General: The graph of 𝑦 = 𝑥 2 + 3 is the graph of y  x 2 is moved along the y-axis…
Up k units if: _____________.
Down k units if: ___________.
2
Investigate 𝑦 = 𝑎𝑥
1
For the following equations 𝑦 = 𝑥 2
𝑦 = −𝑥 2
𝑦 = 2𝑥 2
𝑦 = 𝑥2,
2
1. Complete the table of values.
2. Plot the points on the grid using different colors for each equation.
3. For each equation, complete the chart below the grid.
𝑦 = −𝑥 2
𝒚 = 𝒙𝟐
x
-2
-1
0
y
Mathematics 20-2
x
-2
-1
0
𝑦 = 2𝑥 2
y
x
-2
-1
0
𝑦=
y
Quadratic Functions and Equations
x
-2
-1
0
1 2
𝑥
2
y
Page 54 of 96
1
2
1
2
𝒚 = 𝒙𝟐
1
2
𝑦 = −𝑥 2
1
2
𝑦 = 2𝑥 2
𝑦=
1 2
𝑥
2
Vertex
Maximum or
minimum value
y-intercept
x-intercept(s)
Equation of the
Axis of Symmetry
Domain
Range
In General: The graph of y  ax 2 is the graph of y  x 2 is stretched.
In General:
If a  0 (negative), then the graph opens down.
If 𝑎 > 0 (positive), then the graph opens up.
A ball is thrown into the air from the balcony of an apartment building and falls to the ground.
The height y (in meters) of the ball relative to the ground x seconds after being thrown is
shown by the following quadratic function:
Mathematics 20-2
Quadratic Functions and Equations
Page 55 of 96
1. What is the approximate maximum height of the ball?
2. What is the y-intercept?
3. What is the height of the balcony? How does this height compare to the y-intercept?
4. How long does it take for the ball to hit the ground?
5. What is the x-intercept? How does this compare to the length of time it takes for the ball to
hit the ground?
6. What is height of the ball after 5 seconds? It is going up or down?
7. The point at which the ball changes direction is called __________________?
Mathematics 20-2
Quadratic Functions and Equations
Page 56 of 96
M20-2 Lesson 2 Investigate Vertex Form Key
Investigate 𝑦 = (𝑥 − ℎ)2
For the following equations 𝑦 = 𝑥 2
𝑦 = (𝑥 − 2)2
𝑦 = (𝑥 + 1)2 ,
4. Complete the table of values.
5. Plot the points on the grid using different colors for each equation.
6. For each equation, complete the chart below the grid.
𝒚 = (𝒙 − 𝟐)𝟐
x
y
0
1
2
3
4
𝒚 = 𝒙𝟐
x
-2
-1
0
1
2
y
𝒚 = 𝒙𝟐
Vertex
Maximum or
minimum value
y-intercept
x-intercept(s)
Equation of the
Axis of Symmetry
Domain
Range
(0, 0)
min = 0
𝒚 = (𝒙 + 𝟏)𝟐
x
y
-3
4
-2
1
-1
0
0
1
1
4
𝒚 = (𝒙 − 𝟐)𝟐
(0, 2)
min = 0
𝒚 = (𝒙 + 𝟏)𝟐
(0, -1)
min = 0
0
0
x=0
4
0
x=2
1
0
x = -1
x∈R
y≥0
x∈R
y≥0
x∈R
y≥0
In General: The graph of y  ( x  h) 2 is the graph of y  x 2 is moved along the x-axis…
Left h units if: _____ h < 0________. Right h units if: ______ h > 0_____.
Investigate 𝑦 = 𝑥 2 + 𝑘
For the following equations 𝑦 = 𝑥 2
𝑦 = 𝑥2 + 3
𝑦 = 𝑥 2 − 2,
4. Complete the table of values.
5. Plot the points on the grid using different colors for each equation.
Mathematics 20-2
Quadratic Functions and Equations
Page 57 of 96
6. For each equation, complete the chart below the grid.
𝑦 = 𝑥2 + 3
x
y
0
1
2
3
4
𝒚 = 𝒙𝟐
x
-2
-1
0
1
2
y
𝒚 = 𝒙𝟐
Vertex
Maximum or
minimum value
y-intercept
x-intercept(s)
Equation of the
Axis of Symmetry
Domain
Range
(0, 0)
min = 0
𝑦 = 𝑥2 − 2
x
y
-3
5
-2
0
-1
-1
0
-2
1
-1
𝑦 = 𝑥2 + 3
(0, 2)
min = 3
𝑦 = 𝑥2 − 2
(0, -1)
min = -2
0
0
x=0
3
0
x=0
-2
0
x=0
x∈R
y≥0
x∈R
y≥3
x∈R
y ≥ -2
In General: The graph of 𝑦 = 𝑥 2 + 3 is the graph of y  x 2 is moved along the y-axis…
Up k units if: ______ h > 0_______.
Down k units if: ____ h < 0_______.
Mathematics 20-2
Quadratic Functions and Equations
Page 58 of 96
Investigate 𝑦 = 𝑎𝑥 2
1
For the following equations 𝑦 = 𝑥 2
𝑦 = −𝑥 2
𝑦 = 2𝑥 2
𝑦 = 2 𝑥2,
4. Complete the table of values.
5. Plot the points on the grid using different colors for each equation.
6. For each equation, complete the chart below the grid.
𝑦 = −𝑥 2
𝒚 = 𝒙𝟐
x
-2
-1
0
1
2
y
x
-2
-1
0
1
2
𝒚 = 𝒙𝟐
𝑦 = 2𝑥 2
y
x
-2
-1
0
1
2
𝑦 = −𝑥 2
1 2
𝑥
2
𝑦=
y
𝑦 = 2𝑥 2
Vertex
(0, 0)
(0, 0)
(0, 0)
Maximum or
min = 0
min = 0
min = 0
minimum value
y-intercept
0
0
0
x-intercept(s)
0
0
0
x=0
x=0
x=0
Equation of the
Axis of
Symmetry
Domain
x∈R
x∈R
x∈R
Range
y≥0
y≤0
y≥0
2
2
In General: The graph of y  ax is the graph of y  x is stretched.
x
-2
-1
0
1
2
𝑦=
y
2
0.5
0
0.5
2
1 2
𝑥
2
(0, 0)
min = 0
0
0
x=0
x∈R
y≥0
In General: If a  0 (negative), then the graph opens down.
If 𝑎 > 0 (positive), then the graph opens up.
Mathematics 20-2
Quadratic Functions and Equations
Page 59 of 96
A ball is thrown into the air from the balcony of an apartment building and falls to the ground.
The height y (in meters) of the ball relative to the ground x seconds after being thrown is
shown by the following quadratic function:
8. What is the approximate maximum height of the ball? 155 m
9. What is the y-intercept? y = 100
10. What is the height of the balcony? 100 m How does this height compare to the y-intercept?
The height of the balcony is the y-intercept.
11. How long does it take for the ball to hit the ground? 20 s
12. What is the x-intercept? 20 How does this compare to the length of time it takes for the ball
to hit the ground? The time it takes the ball to hit the ground is the x-intercept.
13. What is height of the ball after 5 seconds? 150 s It is going up or down? up
14. The point at which the ball changes direction is called ______the vertex____________?
Mathematics 20-2
Quadratic Functions and Equations
Page 60 of 96
M20-2 Lesson 2 Jig Saw Applications
Jig-Saw Problem - Circles
The equation shows the height of a kicked soccer ball, h(d) meters as a function of the
horizontal distance, d meters, until it first hits the ground.
h(d )  0.025(d  20) 2  10
a) Graph the function. Include your window choices. Label the axes with height and distance.
b) What is the maximum height of the ball? Label this point on the graph.
c) What is the horizontal distance of the ball from the kicker when it reaches its maximum
height? What characteristic of a parabola does this number represent?
d) How far does the ball travel horizontally from when it is kicked until it hits the ground? Label
this point on the graph. What is this point called?
e) What is the height of the ball when it is 10 m horizontally from the kicker? Label this point
on the graph. How did you find this point?
f) Would an opposing player positioned under the path of the ball 34 m from the kicker be able
to head the ball? Label this point on the graph. Explain how you determined this.
g) State the domain and range for the path of the soccer ball.
Mathematics 20-2
Quadratic Functions and Equations
Page 61 of 96
Jig-Saw Problem - Triangles
The sum of two numbers is 60. Their product is a maximum. Determine the numbers.
Mathematics 20-2
Quadratic Functions and Equations
Page 62 of 96
Jig-Saw Problem - Squares
A rectangular lot is bordered on one side by a stream and on the other three sides by
600 m of fencing. The area of the lot is a maximum. Determine the area.
Mathematics 20-2
Quadratic Functions and Equations
Page 63 of 96
Jig-Saw Problem - Stars
A student who wishes to use a computer lab at a local library must buy a membership.
The library charges $20 for membership. Four hundred students purchase the
membership. The library estimates that for every $4 increase in the membership fee, 40
fewer students will become members. What membership fee will provide the maximum
revenue for the library?
Mathematics 20-2
Quadratic Functions and Equations
Page 64 of 96
M20-2 Lesson 2 Jig Saw Applications Key
Mathematics 20-2
Quadratic Functions and Equations
Page 65 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 66 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 67 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 68 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 69 of 96
M20-2 Lesson 2 Jig Saw Equation Forms
Jig-Saw Problem - Circles
Sketch a graph of the following quadratic function and complete the table provided.
𝑦 = 𝑥 2 + 2𝑥 − 8
𝑦 = 𝑥 2 + 2𝑥 − 8
Vertex
Maximum or minimum
value
y-intercept
x-intercept(s)
Equation of the Axis of
Symmetry
Domain
Range
Mathematics 20-2
Quadratic Functions and Equations
Page 70 of 96
Jig-Saw Problem - Triangles
Sketch a graph of the following quadratic function and complete the table provided.
𝑦 = (𝑥 + 1)2 − 9
𝑦 = (𝑥 + 1)2 − 9
Vertex
Maximum or minimum
value
y-intercept
x-intercept(s)
Equation of the Axis of
Symmetry
Domain
Range
Mathematics 20-2
Quadratic Functions and Equations
Page 71 of 96
Jig-Saw Problem - Squares
Sketch a graph of the following quadratic function and complete the table provided.
𝑦 = (𝑥 + 4)(𝑥 − 2)
𝑦 = (𝑥 + 4)(𝑥 − 2)
Vertex
Maximum or minimum
value
y-intercept
x-intercept(s)
Equation of the Axis of
Symmetry
Domain
Range
Mathematics 20-2
Quadratic Functions and Equations
Page 72 of 96
Jig-Saw Problem - Stars
Sketch a graph of the following quadratic function and complete the table provided.
x
-5
-4
-1
0
2
3
y
7
0
-9
-8
0
7
Vertex
Maximum or minimum
value
y-intercept
x-intercept(s)
Equation of the Axis of
Symmetry
Domain
Range
Mathematics 20-2
Quadratic Functions and Equations
Page 73 of 96
M20-2 Lesson 2 Jig Saw Equation Forms Key
Jig-Saw Problem - Circles
Sketch a graph of the following quadratic function and complete the table provided.
𝑦 = 𝑥 2 + 2𝑥 − 8
y = x^2+2x-8

y




x















𝑦 = 𝑥 2 + 2𝑥 − 8
(-1, -9)
Vertex
Maximum or minimum
value
Minimum y = -9
y-intercept
y = -8
x-intercept(s)
x = -4, 2
Equation of the Axis of
Symmetry
Mathematics 20-2
x = -1
Domain
𝑥∈𝑅
Range
𝑦 ≥ −9
Quadratic Functions and Equations
Page 74 of 96
Jig-Saw Problem - Triangles
Sketch a graph of the following quadratic function and complete the table provided.
𝑦 = (𝑥 + 1)2 − 9
y = x^2+2x-8

y




x















𝑦 = (𝑥 + 1)2 − 9
Vertex
(-1, -9)
Maximum or minimum
value
y-intercept
y = -8
x-intercept(s)
x = -4, 2
Equation of the Axis of
Symmetry
x = -1
Domain
𝑥∈𝑅
Range
Mathematics 20-2
Minimum y = -9
𝑦 ≥ −9
Quadratic Functions and Equations
Page 75 of 96
Jig-Saw Problem - Squares
Sketch a graph of the following quadratic function and complete the table provided.
𝑦 = (𝑥 + 4)(𝑥 − 2)
y = x^2+2x-8

y




x















𝑦 = (𝑥 + 4)(𝑥 − 2)
Vertex
(-1, -9)
Maximum or minimum
value
y-intercept
y = -8
x-intercept(s)
x = -4, 2
Equation of the Axis of
Symmetry
x = -1
Domain
𝑥∈𝑅
Range
Mathematics 20-2
Minimum y = -9
𝑦 ≥ −9
Quadratic Functions and Equations
Page 76 of 96
Jig-Saw Problem - Stars
Sketch a graph of the following quadratic function and complete the table provided.
y = x^2+2x-8

y



x
-5
-4
-1
0
2
3
y
7
0
-9
-8
0
7

x















Vertex
(-1, -9)
Maximum or minimum
value
y-intercept
y = -8
x-intercept(s)
x = -4, 2
Equation of the Axis of
Symmetry
x = -1
Domain
𝑥∈𝑅
Range
Mathematics 20-2
Minimum y = -9
𝑦 ≥ −9
Quadratic Functions and Equations
Page 77 of 96
M20-2 Lesson 2 Matching Activity
Matching 1
Vertex (2, 1)
Mathematics 20-2
Minimum Value at -2
Range 𝑦 ≥ −1
Quadratic Functions and Equations
Page 78 of 96
Domain 𝑥 ∈ 𝑅
Range 𝑦 ≥ 3
Equation for axis of
symmetry is
𝑥 = −5
Opens down
Vertex (-3, -2)
Range 𝑦 ≤ −5
Mathematics 20-2
Quadratic Functions and Equations
Page 79 of 96
Matching 2
𝑦 = (𝑥 + 3)2
𝑦 = 𝑥2 + 3
𝑦 = −2𝑥 2
𝑦 = 2𝑥 2
𝑦 = (𝑥 − 2)2 + 1
𝑦 = (𝑥 + 2)2 − 1
Mathematics 20-2
Quadratic Functions and Equations
Page 80 of 96
𝑦 = (𝑥 − 5)2 + 3
𝑦 = −(𝑥 + 3)2 − 2
𝑦 = −𝑥 2 − 5
x-intercepts are
(-1, 0) & (-3, 0)
Vertex (-2, -1)
x-intercepts are
(-1, 0) & (-3, 0)
Vertex (-2, -1)
Mathematics 20-2
y-intercept is 1
Quadratic Functions and Equations
Page 81 of 96
M20-2 Lesson 2 Matching Activity Key
This activity is designed to reaffirm students’ knowledge of intercepts, vertex,
maximums/minimums, domain, and range. Students work in small groups and match a
quadratic function to a corresponding graph or vocabulary word(s). Teachers will need to
photocopy Matching 1 and Matching 2 (in different colors) and cut each square out. Each
group will receive 24 squares and match them accordingly.
Key
𝑦 = (𝑥 + 3)2
𝑦 = 𝑥2 + 3
𝑦 = −2𝑥 2
𝑦 = 2𝑥 2
𝑦 = (𝑥 − 2)2 + 1
↓
Vertex (2, 1)
𝑦 = (𝑥 + 2)2 − 1
↓
Minimum Value at -2
Range 𝑦 ≥ −1
𝑦 = −(𝑥 + 3)2 − 2
↓
Opens down
Vertex (-3, -2)
𝑦 = −𝑥 2 − 5
↓
Range 𝑦 ≤ −5
𝑦 = (𝑥 − 5)2 + 3
↓
Domain 𝑥 ∈ 𝑅
Range 𝑦 ≥ 3
𝑦 = (𝑥 + 5)2 + 3
↓
Equation for axis of symmetry is
𝑥 = −5
x-intercepts are (-1, 0) & (-3, 0)
Vertex (-2, -1)
y-intercept is 1
Mathematics 20-2
Quadratic Functions and Equations
Page 82 of 96
M20-2 Lesson 3 Rectangles and Algebra Tiles Completion Assignment
Rectangles and Algebra Tiles
Polynomial
Sketch of
Rectangle
Grid
Factors
2x2+3x+1
x
+3
3x +2
3x2 2x
9x 6
(x+3)(x+5)
x
+3
Mathematics 20-2
2x +3
2x2 3x
6x 9
Quadratic Functions and Equations
Page 83 of 96
3x2+13x+4
(6x+3)(x+2)
5x2+8x+3
8x2+18x+9
Mathematics 20-2
Quadratic Functions and Equations
Page 84 of 96
M20-2 Lesson 3 Factoring Review Examples
Factoring Trinomials
Example: 6x2 + 14x - 12
2(3x2 + 7x – 6)
Factor Pictorially
Mathematics 20-2
GCF First!
Factor Symbolically
Quadratic Functions and Equations
Page 85 of 96
Difference of Squares
Example: 27x2 - 75
2(9x2 – 25)
Factor Pictorially
Mathematics 20-2
GCF First!
Factor Symbolically
Quadratic Functions and Equations
Page 86 of 96
M20-2 Lesson 3 Factoring Review Examples Sample Solutions
Mathematics 20-2
Quadratic Functions and Equations
Page 87 of 96
Mathematics 20-2
Quadratic Functions and Equations
Page 88 of 96
M20-2 Lesson 3 Solving Using the Quadratic Formula
In General: The roots (solutions) of the quadratic equation ax2 + bx + c = 0 can be found
using the Quadratic Formula: x =
-b ± b2 - 4ac
2a
Problem 1: Solve the quadratic equation x2 + 5x - 14 = 0 using the quadratic formula.
Problem 2: Solve the quadratic equation x2 - 2x - 15 = 0 using the quadratic formula.
Problem 3: Solve the quadratic equation 2x 2 + 5x - 12 = 0 using the quadratic formula.
Problem 4: Solve the quadratic equation 3x2 + 5x - 2 = 0 using the quadratic formula.
Mathematics 20-2
Quadratic Functions and Equations
Page 89 of 96
Recall:
An entire radical is changed into a mixed radical by factoring the largest perfect
square out of the radicand.
Problem 5: Simplify the following by changing the entire radicals into mixed radicals.
a)
32
6
b)
2+ 8
2
c)
3 - 12
2
d)
9 - 27
6
Problem 6: Solve the quadratic equation 3x 2 + 5x - 11 = 0 using the quadratic formula.
Problem 7: Determine the solutions using the Quadratic Formula.
a) 4x 2 - 12x + 9 = 0
Mathematics 20-2
b) 2t 2 - 11t + 15 = 0
Quadratic Functions and Equations
Page 90 of 96
Problem 8: The height of the Peace Tower in Ottawa is 90 m. If an object is thrown downward
at 5 m/s from this height, the time, t seconds, the object takes to reach the ground
can be found by solving the equation -4.9t 2 - 5t + 90 = 0 . Determine the time, to
the nearest tenth of a second, for the object to hit the ground?
Problem 9: Solve: 15x2 - 37x + 20 = 0.
Mathematics 20-2
Quadratic Functions and Equations
Page 91 of 96
M20-2 Lesson 3 Solving Using the Quadratic Formula Key
In General: The roots (solutions) of the quadratic equation ax2 + bx + c = 0 can be found
-b ± b2 - 4ac
using the Quadratic Formula: x =
2a
Problem 1: Solve the quadratic equation x2 + 5x - 14 = 0 using the quadratic formula.
a = 1, b = 5, c = -14
x=
( ) ( 5)
- 5 ±
2
( )(
- 4 1 -14
()
21
) = -5 ±
25 + 56 -5 ± 81 -5 ± 9
=
=
2
2
2
x = -7, 2
Check
( -7)
2
( )
(2)
+ 5 -7 - 14 = 0
0=0
2
()
+ 5 2 - 14 = 0
0=0
Problem 2: Solve the quadratic equation x2 - 2x - 15 = 0 using the quadratic formula.
a = 1, b = -2, c = -15
x=
( ) ( -2) - 4 (1) ( -15) = 2 ±
2 (1)
2
- -2 ±
4 + 60 2 ± 64 2 ± 8
=
=
2
2
2
x = -3, 5
Check
( -3)
2
( )
(5)
- 2 -3 - 15 = 0
0=0
2
()
- 2 5 -15 = 0
0=0
Problem 3: Solve the quadratic equation 2x 2 + 5x - 12 = 0 using the quadratic formula.
a = 2, b = 5, c = -12
x=
( ) ( 5)
- 5 ±
2
( )(
) = -5 ±
- 4 2 -12
()
2 2
25 + 96 -5 ± 121 -5 ± 11
=
=
4
4
4
x = -4, 3/2
Check
( )
2
2
( )
2 -4 + 5 -4 - 12 = 0
0=0
Mathematics 20-2
æ 3ö
æ 3ö
2 ç ÷ + 5 ç ÷ - 12 = 0
è 2ø
è 2ø
0=0
Quadratic Functions and Equations
Page 92 of 96
Problem 4: Solve the quadratic equation 3x2 + 5x - 2 = 0 using the quadratic formula.
a = 3, b = 5, c = -2
x=
( ) (5) - 4 (3) ( -2) = -5 ±
2 ( 3)
2
- 5 ±
25 + 24 -5 ± 49 -5 ± 7
=
=
6
6
6
x = -2, 1/3
Check
( )
2
æ 1ö
æ 1ö
3ç ÷ + 5ç ÷ - 2 = 0
è 3ø
è 3ø
0=0
( )
2
3 -2 + 5 -2 - 2 = 0
0=0
Problem 5: Simplify the following by changing the entire radicals into mixed radicals.
32 4 2 2 2
=
=
6
6
3
a)
c)
3 - 12 3 - 2 3
=
2
2
b)
2+ 8 2+2 2
=
= 1+ 2
2
2
d)
9 - 27 9 - 3 3 3 - 3
=
=
6
6
2
Problem 6: Solve the quadratic equation 3x 2 + 5x - 11 = 0 using the quadratic formula.
a = 3, b = 5, c = -11
x=
( ) ( 5)
- 5 ±
2
( ) ( ) = -5 ±
- 4 3 -11
()
23
25 + 132 -5 ± 157
=
6
6
Problem 7: Determine the solutions using the Quadratic Formula.
a) 4x 2 - 12x + 9 = 0
a = 4, b = -12, c = 9
x=
(
) ( -12)
2 ( 4)
- -12 ±
2
( ) ( ) = 12 ±
-4 4 9
144 - 144 12 ± 0
=
8
8
x = 3/2
Check
2
æ 3ö
æ 3ö
4 ç ÷ - 12 ç ÷ + 9 = 0
è 2ø
è 2ø
0=0
b) 2t 2 - 11t + 15 = 0
a = 2, b = -11, c = 15
Mathematics 20-2
Quadratic Functions and Equations
Page 93 of 96
x=
( ) ( -11) - 4 (2) (15) = 11±
2 ( 2)
2
- -11 ±
121- 120 11± 1
=
4
4
x = 5/2, 3
Check
2
æ 5ö
æ 5ö
2 ç ÷ - 11ç ÷ + 15 = 0
è 2ø
è 2ø
0=0
()
()
2
2 3 - 11 3 + 15 = 0
0=0
Problem 8: The height of the Peace Tower in Ottawa is 90 m. If an object is thrown downward
at 5 m/s from this height, the time, t seconds, the object takes to reach the ground
can be found by solving the equation -4.9t 2 - 5t + 90 = 0 . Determine the time, to
the nearest tenth of a second, for the object to hit the ground?
-4.9t 2 - 5t + 90 = 0
a = -4.9, b = -5, c = 90
x=
( ) ( -5) - 4 ( -4.9) (90) = 5 ±
2 ( -4.9)
2
- -5 ±
25 + 1764 5 ± 1789
=
-9.8
-9.8
x = -4.826 …, 3.805 …
Since x > 0, x = 3.8 (10th)
Problem 9: Solve: 15x2 - 37x + 20 = 0.
a = 15, b = -37, c = 20
x=
(
) ( -37) - 4 (15) (20) = 37 ±
2 (15)
- -37 ±
2
1369 - 1200 37 ± 169 37 ± 13
=
=
30
30
30
x = 4/5, 5/3
Check
2
æ 4ö
æ 4ö
15 ç ÷ - 37 ç ÷ + 20 = 0
è 5ø
è 5ø
0=0
Mathematics 20-2
2
æ 5ö
æ 5ö
15 ç ÷ - 37 ç ÷ + 20 = 0
è 3ø
è 3ø
0=0
Quadratic Functions and Equations
Page 94 of 96
M20-2 Lesson 3 Solving Equations Questions
Group 1
a) x2 - 6x + 8 = 0
b) 2t2 - 6t + 5 = 0
c) 3x2 + 15x = 0
d) x2 - 36 = 0
Group 6
a) x2 - x - 6 = 0
b) 4x2 + 31x - 8 = 0
c) 2x2 - 22x = 0
d) x2 - 4 = 0
Group 2
a) x2 - x - 6 = 0
b) 4x2 + 31x - 8 = 0
c) 2x2 - 22x = 0
d) x2 - 4 = 0
Group 7
a) x2 + 12x + 32 = 0
b) 2x2 - 3x - 9 = 0
c) 3x2 + 27x = 0
d) x2 - 121 = 0
Group 3
a) t2 + 4t - 12 = 0
b) 4t2 + 9t - 9 = 0
c) 4x2 - 8x = 0
d) x2 - 144 = 0
Group 8
a) x2 + 8x - 180 = 0
b) 3x2 - 5x + 2 = 0
c) 8x2 - 56x = 0
d) x2 - 100 = 0
Group 4
a) a2 + 4a - 21 = 0
b) 2t2 + 5t - 12 = 0
c) -5x2 + 25x = 0
d) x2 - 25 = 0
Group 9
a) x2 - 3x + 2 = 0
b) 3x2 - 7x + 2 = 0
c) -6x2 + 12x = 0
d) x2 - 1 = 0
Group 5
a) x2 + 2x + 1 = 0
b) 2x2 - 11x - 6 = 0
c) -7x2 - 21x = 0
d) x2 - 49 = 0
Mathematics 20-2
Quadratic Functions and Equations
Page 95 of 96
M20-2 Lesson 3 Solving Equations Questions Key
Group 1
a) 2, 4
b) 1.5, 0.5
c) 0, -5
d) +6, -6
Group 6
a) 3, -5
b) -1, 2.5
c) 0, 7
d) +3, -3
Group 2
a) 3, -2
b) - 8, 0.25
c) 0, 11
d) +2, -2
Group 7
a) -4, -8
b) 3, -1.5
c) 0, -9
d) +11, -11
Group 3
a) -6, 2
b) -3, 0.75
c) 0, 2
d) +12, -12
Group 8
a) 10, -18
b) 2/3, 1
c) 0, 7
d) +10, -10
Group 4
a) -7, 3
b) -4, 1.5
c) 0, 5
d) +5, -5
Group 9
a) 2, 1
b) 1/3, 2
c) 0, 2
d) +1, -1
Group 5
a) -1
b) -0.5, 6
c) 0, -3
d) +7, -7
Mathematics 20-2
Quadratic Functions and Equations
Page 96 of 96