MATHEMATICS 20-1
Quadratic Functions and Equations
High School collaborative venture with
Edmonton Christian, Harry Ainlay, J. Percy Page, Jasper
Place, Millwoods Christian, Ross Sheppard and W. P. Wagner
Edm Christian High: Aaron Trimble
Harry Ainlay: Ben Luchkow
Harry Ainlay: Darwin Holt
Harry Ainlay: Lareina Rezewski
Harry Ainlay: Mike Shrimpton
J. Percy Page: Debbie Younger
Jasper Place: Matt Kates
Jasper Place: Sue Dvorack
Millwoods Christian: Patrick Ypma
Ross Sheppard: Patricia Elder
W. P. Wagner: Amber Steinhauer
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 – 2011
Mathematics 20-1
Quadratic Functions and Equations
Page 2 of 50
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 Task (on a separate page which could be photocopied & handed out to
students)
Mars Rover Bouncing Airbags
Teacher Notes for Transfer Task
Transfer Task
Glossary/Rubric
Possible Solution
7
10
17
19
STAGE 3 LEARNING PLANS
Lesson #1
Characteristics of Quadratic Functions
24
Lesson #2
Transformations
28
Lesson #3
Factoring
32
Lesson #4
Solving/Graphing Quadratics by Factoring or Quadratic Formula
35
Lesson #5
Discriminant
38
Lesson #6
Completing the Square
41
Lesson #7
Determining Quadratic Equations
45
Lesson #8
Applications of Quadratics
48
Mathematics 20-1
Quadratic Functions and Equations
Page 3 of 50
Mathematics 20-1
Quadratic Functions and Equations
STAGE 1
Desired Results
Big Idea:
Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and
business.
Enduring Understandings:
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Students will understand …



The basic shape and characteristics of the graph of a quadratic function.
Quadratic functions and equations relate to real life situations.
Solving an equation means to find the root(s).
Essential Questions:
To be given early in the unit:
 Find two examples of real-life situations that could be represented by a quadratic
function.
 What factors must you consider when deciding
how much to charge for a product?
Implementation note:
To be given before studying vertex form:
 Get as close as you can to the equation of a
parabola with vertex (-2,5).
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Other:
 Given the graph of the parabola shown, find the equation.
 Discuss the similarities and differences between linear and quadratic functions.
 Compare and contrast y = (x + 2) to y = (x + 2)(x – 3)?
 How does this graph describe the revenue of a school dance?
 In what situations is one method of solving a quadratic equation preferable to
another?
 Given y = a(x - p)2 + q and the values of a and q, how can you determine the
number of x-intercepts on the corresponding graph?
 Compare and contrast y = x 2 and y = 2x.
 How can algebra tiles be used to model completing the square?
Mathematics 20-1
Quadratic Functions and Equations
Page 4 of 50
Knowledge:
Enduring
Understanding
Specific
Outcomes
Students will know…
Students will understand…
 The basic shape and
characteristics of the
graph of a quadratic
function.
Description of
Knowledge
*RF3
RF4
Students will understand…
 Quadratic functions
and equations relate
to real life situations.
RF4
RF5
Students will understand…
 Solving an equation
means to find the
root(s).
RF1
RF5

that an equation in the form y = x2 is a
quadratic function where the graph is a
parabola
 the effects of changing each parameter in the
function y = a(x - p)2 + q
 that (p, q) is the vertex
 that x = p is the axis of symmetry
 that a determines the direction of opening of
the parabola
 that the x-intercept is found where y = 0 and
the y-intercept is found where x = 0
 the domain is the set of permissible values of
x in the function and range is all the possible
values of y in the function
Students will know…

what a, p, q means in the context of a
problem
 there are many methods available to solve
problems involving quadratics
Students will know…


8888
I*RF =
the zeros of a function correspond to the xintercepts of a graph
the discriminant (b2- 4ac) can be used to
determine the nature of the roots
Relations and Functions
Mathematics 20-1
Quadratic Functions and Equations
Page 5 of 50
Skills:
Enduring
Understanding
Specific
Outcomes
Students will be able to…
Students will understand…
 The basic shape and
characteristics of the
graph of a quadratic
function.
Description of
Skills
*RF3
RF4




Students will understand…
 Quadratic functions and
equations relate to real
life situations.
RF4
RF5
Students will understand…
 Solving an equation
means to find the
root(s).
RF1
RF5
Students will be able to…

solve a quadratic equation of the form
ax2 + bx + c = 0 algebraically and graphically
 determine an appropriate domain and range
 use the discriminate to determine the nature
and number of roots
 analyze a problem and then determine and
solve the quadratic equation
Students will be able to…









I*RF
identify/determine the coordinates of the vertex
sketch the graph of y = a(x - p)2 + q. with or
without technology
write a quadratic function in the form y = a(x p)2 + q. or y = ax2 + bx + c given a graph or a
set of characteristics of a graph
change the form of a quadratic equation by
completing the square or expanding
solve a quadratic equation of the form
ax2 + bx + c = 0 by:
determining square roots
factoring
completing the square
applying the quadratic formula
graphing its corresponding function
verifying the solution
identifying and correcting errors in a solution to
a quadratic equation
analyzing a problem and then determining and
solving the quadratic equation
= 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 20-1
Quadratic Functions and Equations
Page 6 of 50
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Mars Rover Bouncing Airbags
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts relating to
quadratic equations and functions. Photocopy-ready version of the transfer task and
rubric are included in this section.
Each student will . . .
 create data for each participant
 analyze data and plot graphs of the data
 interpret the graphs
 identify restricted domain and range for each simulation
 identify intercepts and vertices and explain their significance
Part A: Students will bounce a ball in front of a scale (meter stick) and record the initial
coordinates, the vertex and the coordinates of the endpoint. Students may film the
bouncing of the ball and analyze video to get more accurate measurements.
Part B: Teachers should encourage students to use a variety of different methods
including completing the square, factoring, using symmetry or quadratic formula.
Part C: Any range of appropriate shifts should be accepted.
Part D: A c value has been added (y = ax2 + bx + c) indicating that the initial point of
impact is above the surface (ie. rock, landing pad, etc.) or the initial point of impact has
been moved backwards
Part E: Any reasonable answer accepted.
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 20-1
Quadratic Functions and Equations
Page 7 of 50
Teacher Notes for Mars Rover Bouncing Airbags Transfer Task
Glossary/Memory Aides
axis of symmetry – An axis of symmetry is a line about which a figure is symmetrical. If a
figure can be folded such that the two parts exactly match, the fold line would be an axis
of symmetry.
direction of opening –
domain – All the x-values of the points of the graph of the function
maximum value – Greatest y-value of the graph
minimum value – Lowest y-value of the graph
range – All the y-values of the points of the graph of the function
vertex – The maximum/minimum point of a parabola
x-intercept – The x-coordinate of the intersection point of a graph and the horizontal axis
y-intercept – The y-coordinate of the intersection point of a graph and the vertical axis
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-1
Quadratic Functions and Equations
Page 8 of 50
The following applet includes most of the glossary terms. Understanding the display will
help students remember quadratic terms.
Source: http://www.ronblond.com/MathGlossary/Division04/Quad01/
Mathematics 20-1
Quadratic Functions and Equations
Page 9 of 50
Mars Rover Bouncing Airbags - Student Assessment Task
You are employed by the Canadian Space Agency and are involved in the testing of
new high-tech polymer rubber. CSA is hoping to sell the idea to NASA for the Mars
Rover Mission planned for 2019. You will have to do some research before you will be
able to present your findings to NASA.
The high-tech polymers will be used in the airbags, which are released from the rover
when it reaches the surface of Mars. See the photos below to get an idea of what it
looks like:
Mars Rover Bouncing
Mars Rover Landed
http://marsrover.nasa.gov/spotlight/images/rocknroll-image02_br.jpg
http://marsrover.nasa.gov/spotlight/images/rocknroll-image03_br.jpg
Role: Canadian Space Agency Tester for New Polymer Rubbers
Audience: NASA
Format: Present using a poster, PowerPoint or booklet,
Topic: You are presenting findings from Super-Polymer Rubber research to sell this
product to NASA for their 2019 mission to Mars.
Materials:
1. ball
2. metre sticks or metre tape measures
Suggestion: Complete in groups of 3.
Mars Rover Bouncing Airbags
Height
Initial
Impact
Point
Horizontal distance travelled between first & second bounce
Your experimental research is comprised of the following parts:
Part A

In a simple ball-bouncing experiment, measure the distance between the first
and second bounces, as well as the maximum height of the bounce. Write an
equation in y = a(x – p)2 + q form and then in y = ax2 + bx + c form which
describes this experiment. Use the space below to sketch and label your
graph
Let x represent the horizontal distance,
and y represent the vertical height.
Distance between first & second
bounce: ________cm
Maximum height: ________cm
In y = a(x – p)2 + q form:
_______________________
In y = ax2 + bx + c form:
_______________________

Analyze the function you created by stating the following:
Vertex _________________________ Axis of symmetry ________________
Direction of opening ______________
Domain ________________________
Range _________________
x-intercepts _____________________
y-intercept _________________
Part B
1. You need to see how 3 different polymers would react on Mars, where the
gravity is 38% of that on Earth. A lab is created where this gravity is simulated
and the results show the following for the high-tech polymers.
Polymer A: y = -0.005x2 + 0.2x
Polymer B: y = -0.0625x2 + 0.5x
Polymer C: y = -0.375x2 + 3.75x
Using algebra, determine which polymer will travel the furthest
between the first bounce (initial impact point) and the second bounce,
and which polymer attains the maximum height. Show your work in
the space below.
Part C
1. On a satellite image, it is noticed that there is a 1.5 m high landform, 38 m from
the first impact point. Choose one polymer and determine where the first
impact point needs to be moved to, in order for that polymer to clear the object
between the first and second bounce. Sketch the original graph and the new
one you have determined on the grid below. What is the new domain of the
graph?
2. State all possible positions for the first impact point.
Part D
1. In the simulation lab, a new experiment was performed on Polymer C with a
new set of initial conditions. This was the result:
y = -0.375x2 + 3.75x + 6
Explain, in the context of this problem, what initial condition was changed?
Algebraically, determine the horizontal distance travelled by Polymer C
between the first and second bounce to the nearest tenth of a metre.
What changes have there been in the domain and range compared to the first
experiment with Polymer C? You may use your graphing calculator. Answer in
the box below:
Part E
When you approach NASA with your polymers, which polymer might you recommend,
and why? Defend your choice by using the results of the lab simulation testing.
\
http://3.bp.blogspot.com/_56uq77msz_g/SfbYVk2cPvI/AAAAAAAAATQ/i9iEXR9yWRo/s1600-h/mars2-16-09.jpg
Glossary
axis of symmetry – An axis of symmetry is a line about which a figure is symmetrical.
If a figure can be folded such that the two parts exactly match, the fold line would be
an axis of symmetry.
direction of opening –
domain – All the x-values of the points of the graph of the function
maximum value – Greatest y-value of the graph
minimum value – Lowest y-value of the graph
range – All the y-values of the points of the graph of the function
vertex – The maximum/minimum point of a parabola
x-intercept – The x-coordinate of the intersection point of a graph and the horizontal
axis
y-intercept – The y-coordinate of the intersection point of a graph and the vertical axis
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
The following applet includes most of the glossary terms. Understanding the
display will help students remember quadratic terms.
Source: http://www.ronblond.com/MathGlossary/Division04/Quad01/
Assessment
Mathematics 20-1
Quadratic Functions and Equations Rubric
Level
Criteria
Analyze
Quadratic
Functions of the
form
y = a(x - p)2 + q.
(Parts A-D)
Performs
Algebraic
Operations On
Quadratic
Functions In the
Form
ax 2 + bx + c = 0
(Part B & D)
Solve Problems
That Involve
Quadratic
Equations
(Part A,C,D,E)
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
Student is able to
determine all
features of all
required functions
Student is able to
determine most
features of all
required functions
Student is able to
determine some
features of all
required functions
Student is able to
determine few
features of all
required functions
Student is unable
to determine any
features of
quadratic functions
Student is able to
correctly perform a
variety of algebraic
operations
including factoring,
completing the
square, quadratic
formula and/or
symmetry
Student is able to
correctly perform
most of the
algebraic
operations
including factoring,
completing the
square, quadratic
formula and/or
symmetry
Student
demonstrates
limited knowledge
of algebra or
combination of
algebra and use of
graphing calculator
Student only used
graphing calculator
or algebra
contained several
major errors
Student
demonstrated no
significant
progress towards
correct responses
Student derives
equation correctly
using his/her data
Student work
shows up to 3
minor errors
and/or one major
error
Student
demonstrates
correct approach
on Parts A, C & D
but makes
significant errors in
calculations
Student
demonstrates
some knowledge
of quadratics in
their work
No knowledge of
quadratics is
apparent
Student
demonstrates
algebraic steps
and graphs clearly
Students
demonstrate some
algebraic steps
and/or graphs
incomplete
Student
demonstrates poor
communication of
math.
Student’s work
must be legible,
neat and
organized in a
logical manner
Student’s work is
difficult to follow
Student work
illustrates poor
presentation
Student finds all
possible values in
Part C
Student
determines correct
solution in Part D
OR
At least two
solutions are
correctly
demonstrated with
minor errors
accepted
Student provides
logical justification
in Part E
Math
Communication
Presentation
Explains work
clearly with all
details present
and correct
answer where
applicable.
Uses clear and
effective diagrams,
tables, charts and
graphs.
Explains work,
with most details
present and may
or may not have
the correct
answer.
Appropriate but
incomplete use of
diagrams, tables,
charts and graphs.
Possible Solution to Mars Rover Bouncing Airbags
You are employed by the Canadian Space Agency and are involved in the testing of
new high-tech polymer rubber. CSA is hoping to sell the idea to NASA for the Mars
Rover Mission planned for 2019. You will have to do some research before you will be
able to present your findings to NASA.
The high-tech polymers will be used in the airbags, which are released from the rover
when it reaches the surface of Mars. See the photos below to get an idea of what it
looks like:
Mars Rover Bouncing
Mars Rover Landed
http://marsrover.nasa.gov/spotlight/images/rocknroll-image02_br.jpg
http://marsrover.nasa.gov/spotlight/images/rocknroll-image03_br.jpg
Role: Canadian Space Agency Tester for New Polymer Rubbers
Audience: NASA
Format: Present using a poster, PowerPoint, booklet
Topic: You are presenting findings from Super-Polymer Rubber research to sell this
product to NASA for their 2019 mission to Mars.
Materials: A ball and metre stick or tape measure
Suggestion: To be done in groups of 3
Mars Rover Bouncing Airbags
Height
Initial
Impact
Point
Horizontal distance travelled between first & second bounce
Mathematics 20-2
Inductive and Deductive Reasoning
Page 17 of 50
Your experimental research is comprised of the following parts:
Part A (Sample solution, as all student solutions will differ based on their
findings)
1. In a simple ball-bouncing experiment, measure the distance between the first and
second bounces, as well as the maximum height of the bounce. Write an equation
in y = a(x – p)2 + q form and then in y = ax2 + bx + c form which describes this
experiment. Use the space below to sketch and label your graph
Let x represent the horizontal distance,
and y represent the vertical height.
Distance between first & second
bounce: ___32____cm
Maximum height:____64___cm
In y = a(x – p)2 + q form:
__ y = -0.25(x – 16)2 + 16 __
In y = ax2 + bx + c form:
___y = -0.25x2 + 8x _______
Sample solution:
Use the x-intercept to determine a.
y = a ( x - 16)2 + 64
( )
2
0 = a éë 32 - 16ùû + 64
0 = 256a + 64
a =-
1
or -0.25
4
Write the equation in y = a(x – p)2 + q form and then in y = ax2 + bx + c form.
y = -0.25( x - 16)2 + 64
= -0.25( x 2 - 32x + 256) + 64
= -0.25x 2 + 8x
2. Analyze the function you created by stating the following:
Vertex _______(16, 64)___________ Axis of symmetry ______x = 16_____
Direction of opening __downward___
Domain __0 < x < 32_____________ Range ___0 < y < 64_______
x-intercepts ___(0, 0) & (32, 0)_____ y-intercept _______(0, 0)_______
Mathematics 20-2
Inductive and Deductive Reasoning
Page 18 of 50
Part B
1. You need to see how 3 different polymers would react on Mars, where the
gravity is 38% of the Earth. A lab is created where this gravity is simulated and
the results show the following for the high-tech polymers.
Polymer A: y = -0.005x2 + 0.2x
Polymer B: y = -0.0625x2 + 0.5x
Polymer C: y = -0.375x2 + 3.75x
Using algebra, determine which polymer will travel the furthest
between the first and second bounce, and which polymer attains the
maximum height. Show your work in the space below.
Polymer A:
y = -0.005x(x - 40)
0 = -0.005x(x - 40)
x = 0, 40
Factor to determine the x-intercepts
y = -0.005x2 + 0.2x
Complete the square to determine the vertex
2
y = -0.005(x - 40x)
y = -0.005(x2 - 40x + 400 - 400)
y = -0.005(x – 20) 2 + (-0,005)(-400)
y = -0.005(x – 20) 2 + 2
The x-intercepts are (0, 0) and (40, 0), therefore the distance travelled is 40 m.
The vertex is at (20, 2) and therefore maximum height is 2 m.
Polymer B:
x =
-b ± b 2 - 4ac
2a
Use the quadratic formula to determine the x-intercepts
( ) (0.5) - 4 ( -0.0625) (0)
x =
2 ( -0.0625)
- (0.5) ± (0.5)
=
- 0.5 ±
2
2
=
-0.125
- 0.5 ± 0.5
( )
-0.125
x = 0, 8
y = -0.0625(x2 - 8x + 16 - 16)
Complete the square to determine the vertex
y = -0.0625(x – 4) 2 + (-0,0625)(-16)
y = -0.0625(x – 4) 2 + 1
The x-intercept is (8, 0) and distance travelled is 8 m. The vertex is at (4, 1) and
the maximum height is 1 m.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 19 of 50
Polymer C:
y = -0.375x2 + 3.75x
y = -0.375x(x – 10)
0 = -0.375x(x – 10)
Use factored form to determine the x-intercepts
x = 0, 10
y = -0.375(x2 - 10x + 25 - 25)
Complete the square to determine the vertex
y = -0.375(x – 5) 2 + (-0.375)(-25)
y = -0.375(x – 5) 2 + 9.375
The x-intercept is (10, 0) and distance travelled is 10 m. The vertex is at
(5, 9.375) and the maximum height is 9.375 m.
Polymer A travels the furthest (40 m).
Polymer C attains the highest point (9.375 m).
Mathematics 20-2
Inductive and Deductive Reasoning
Page 20 of 50
Part C
1. On a satellite image, it is noticed that there is a 1.5 m high landform, 38 m from
the first impact point. Choose one polymer and determine where the first
impact point needs to be moved to, in order for that polymer to clear the object
between the first and second bounce. Sketch the original graph and the new
one you have determined on the grid below. What is the new domain of the
graph?
Graphs will vary and domains will
vary.
A typical solution for Polymer A
might be to move the vertex to
(38,2), shifting the graph 18 units
right. Thus, the first impact point
would be at (18,0). The domain of
this graph would be
18 m < x < 58 m.
However, moving the initial impact
point anywhere between, but not
including, 10 m and 30 m to the
right would lead to a correct
solution.
Polymer B will not clear.
2. State all possible positions for the first impact
point.
See to the right.
Mathematics 20-1
A typical solution for Polymer C
would be to move the vertex to
(38, 9.375), shifting the graph 33
units right. Thus, the first impact
point would be at (33,0). The
domain of this graph would be
33 m < x < 43 m.
However, moving the initial impact
point anywhere between, but not
including, 28.4 m and 37.6 m to
the right would lead to a correct
solution.
Quadratic Functions and Equations
Page 21 of 50
Part D
1.
In the simulation lab, a new experiment was performed on Polymer C with a
new set of initial conditions. This was the result:
y = -0.375x2 + 3.75x + 6
Explain, in the context of this problem, what initial condition was changed?
Algebraically, determine the horizontal distance travelled by Polymer C
between the first and second bounce to the nearest tenth of a metre.
What changes have there been in the domain and range compared to the first
experiment with Polymer C? You may use their graphing calculator. Answer in
the box below:
Solution 1:
The initial impact point height has changed to 6 m. (Possibly it bounced off an
object in the lab during the experiment)
x =
(
) (3.75) - 4 ( -0.375) (6)
2 ( -0.375)
2
- 3.75 ±
-3.75 ± 23.0625
-0.75
x = 11.4 m, -1.4 m
x = 11.4 m, -1.4 m
=
(
)
New Domain: 0 £ x £ 11.4
(
)
New Range: 0 £ y £ 15.375
Note: This range calculation does not have to be done algebraically, and can
be done using the GDC.
Solution 2:
The landing point was moved 1.4 m to the left./The polymer landed to the left of
the original landing point in the experiment.
x =
(
) (3.75) - 4 ( -0.375) (6)
2 ( -0.375)
2
- 3.75 ±
-3.75 ± 23.0625
-0.75
x = 11.4 m, -1.4 m
=
(
)
New Domain: -1.4 £ x £ 11.4
(
)
New Range: 0 £ y £ 15.375
Note: This range calculation does not have to be done algebraically. You can
get the same result be using the GDC.
Mathematics 20-1
Quadratic Functions and Equations
Page 22 of 50
Part E
When you approach NASA with your polymers, which polymer might you recommend,
and why? Defend your choice by using the results of the lab simulation testing.
Answers will vary.
Mathematics 20-1
Quadratic Functions and Equations
Page 23 of 50
STAGE 3
Learning Plans
Lesson 1
Characteristics of Quadratic Functions
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …


The basic shape and characteristics of the
graph of a quadratic function.




Find two examples of real-life situations that
could be represented by a quadratic function.
Get as close as you can to the equation of a
parabola with vertex (-2,5).
Given the graph of the parabola shown, find
the equation.
Discuss the similarities and differences
between linear and quadratic functions.
Given y = a(x - p)2 + q and the values of a and
q, how can you determine the number of xintercepts on the corresponding graph?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …

that an equation in the form y = x2 is a
quadratic function where the graph is a
parabola
 the effects of changing each
parameter in the function y = a(x - p)2 + q.
 that (p, q) is the vertex
 that x = p is the axis of symmetry
 that a determines the direction of opening
of the parabola
 the domain is the set of permissible values of x
in the function and range is all the possible
values of y in the function


identify/determine the coordinates of the
vertex
sketch the graph of y = a(x – p)2 + q 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 20-1
Quadratic Functions and Equations
Page 24 of 50
Lesson Summary

Students will be able to sketch the graph in the form y = a(x – p)2 + q (with and
without technology) and be able to identify graph characteristics.
Lesson Plan
Hook
If you are looking for a Characteristics of Quadratic Functions video, consider:
http://www.youtube.com/watch?v=1_GW3gNgpNk
Note: The axis of symmetry is calculated using x =
-b ± b 2 - 4ac
.
2a
Lesson Goal
Students will be able to sketch the graph in the form y = a(x – p)2 + q (with and without
technology) and be able to identify graph characteristics.
Lesson
Have students work on quadratic functions (with graphing calculator or applet – see
below), starting with the basic quadratic function and then progressing in complexity;
all in the form y = a(x – p)2 + q. Use y = ax2 and change values of a. Have students
come up with a generalization of how a effects the graph. Then, keeping a constant,
give examples with only the p value changing. Then proceed with changing q.
Identify
 axis of symmetry
 vertex
 max/min value
 domain
 range
 direction of opening
 x-intercepts
 y-intercepts
Mathematics 20-1
Quadratic Functions and Equations
Page 25 of 50
Going Beyond
Have students develop their own understanding of how the equation relates to certain
characteristics of the graph.
Could discuss parameter a as it relates to a horizontal stretch.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 3.1)
Supporting
Use a quadratic function applet:
http://www.ronblond.com/M11/QFA.CSF.APPLET/index.html
http://www.ronblond.com/MathGlossary/Division04/Quad01/
Assessment
A quick quiz could be given at the end of the class. No calculators. Match 2
equations to the correct 2 of 4 graphs drawn beside.
Mathematics 20-1
Quadratic Functions and Equations
Page 26 of 50
Glossary/Memory Aides
axis of symmetry – An axis of symmetry is a line about which a figure is symmetrical.
If a figure can be folded such that the two parts exactly match, the fold line would be
an axis of symmetry.
direction of opening –
domain – All the x-values of the points of the graph of the function
maximum value – Greatest y-value of the graph
minimum value – Lowest y-value of the graph
range – All the y-values of the points of the graph of the function
vertex – The maximum/minimum point of a parabola
x-intercept – The x-coordinate of the intersection point of a graph and the horizontal
axis
y-intercept – The y-coordinate of the intersection point of a graph and the vertical axis
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Note to teachers:
It would be helpful to bring in laptops or go to computer lab to get students using the
(
applet above to draw conclusions and make generalizations about y = a x - p
)
2
+q .
The generalizations are more immediate with the applet than calculator.
One of the applets uses y = a(x – h)2 + k instead of y = a(x – p)2 + q. The slider label
can be changed in the HTML code that displays the applet.
The other applet uses Standard instead of Vertex and General instead of Standard.
The button names can be changed in the HTML code that displays the applet.
Mathematics 20-1
Quadratic Functions and Equations
Page 27 of 50
Lesson 2
Transformations
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …


The basic shape and characteristics of the
graph of a quadratic function.






Find two examples of real-life situations that
could be represented by a quadratic function.
Get as close as you can to the equation of a
parabola with vertex (-2,5).
Given the graph of the parabola shown, find
the equation.
Discuss the similarities and differences
between linear and quadratic functions.
Compare and contrast y = (x + 2) to
y = (x + 2)(x – 3)?
How does this graph describe the revenue of
a school dance?
Given y = a(x - p)2 + q and the values of a and
q, how can you determine the number of xintercepts on the corresponding graph?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …







that an equation in the form y = x2 is a
quadratic function where the graph is a
parabola
the effects of changing each parameter in the
function y = a(x - p)2 + q
that (p, q) is the vertex
that x = p is the axis of symmetry
that a determines the direction of opening of
the parabola
that the x-intercept is found where y = 0 and
the y-intercept is found where x = 0
the domain is the set of permissible values of
x in the function and range is all the possible
values of y in the function



identify/determine the coordinates of the
vertex
sketch the graph of y = a(x – p)2 + q with or
without technology
write a quadratic function in the form
y = a(x – p)2 + q or y = ax2 + bx + c given a
graph or a set of characteristics of a graph
Lesson Summary

Students will translate a quadratic equation both vertically and horizontally as
well as determine reflections about the x-axis.
Mathematics 20-1
Quadratic Functions and Equations
Page 28 of 50
Lesson Plan
Hook
Show students the following clip:
http://www.youtube.com/watch?v=0G00eni0vN0&safety_mode=true&persist_safety_m
ode=1
Learning Goal
Students will translate a quadratic equation both vertically and horizontally as well as
determine reflections about the x-axis.
Activating Prior Knowledge
Review a question from Lesson 1. Have students apply their knowledge of the
characteristics.
Mathematics 20-1
Quadratic Functions and Equations
Page 29 of 50
Lesson
1. Let the students come up with an example of each type of translation
independently. Have students trade their examples with another student and verify
them. Bring a few of the examples up to the board to show the class.
2. The teacher introduces combination example(s):
 describing transformations from equations
 making equations from descriptions
 making an equation from a graph
 describe transformations from a graph
3. The teacher assigns practice questions for students to complete.
4. The teacher provides reflection examples and checks for understanding (orally).
5. The teacher assigns practice questions for students to complete.
Going Beyond
The teacher demonstrates transforming an equation and graph using a coordinate.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 3.1, 3.2, 3.3)
This section will involve jumping around the text.
Supporting
Assessment
Have students start at the origin on a grid. Then teacher will describe a series of
transformations and students will come up with the correct resulting position.
Mathematics 20-1
Quadratic Functions and Equations
Page 30 of 50
Glossary
reflection – The mirror image of a function about an axis or line
transformation – Changing the position, direction or shape of a function
translation – The vertical or horizontal shift of a function
Other
Mathematics 20-1
Quadratic Functions and Equations
Page 31 of 50
Lesson 3
Factoring
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




The basic shape and characteristics of the
graph of a quadratic function.
Quadratic functions and equations relate to
real life situations.
Solving an equation means to find the root(s).






Find two examples of real-life situations that
could be represented by a quadratic function.
What factors must you consider when
deciding how much to charge for a product?
Given the graph of the parabola shown, find
the equation.
Compare and contrast y = (x + 2) to
y = (x + 2)(x – 3)?
How does this graph describe the revenue of
a school dance?
In what situations is one method of solving a
quadratic equation preferable to another?
Given y = a(x - p)2 + q and the values of a and
q, how can you determine the number of xintercepts on the corresponding graph?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …







that an equation in the form y = x2 is a
quadratic function where the graph is a
parabola
that a determines the direction of opening of
the parabola
that the x-intercept is found where y = 0 and
the y-intercept is found where x = 0
the domain is the set of permissible values of
x in the function and range is all the possible
values of y in the function
there are many methods available to solve
problems involving quadratics
the zeros of a function correspond to the xintercepts of a graph
the discriminant (b2- 4ac) can be used to
determine the nature of the roots
Mathematics 20-1









solve a quadratic equation of the form
ax2 + bx + c = 0 algebraically and graphically
use the discriminate to determine the nature
and number of roots
analyze a problem and then determine and
solve the quadratic equation
solve a quadratic equation of the form
ax2 + bx + c = 0 by:
factoring
graphing its corresponding function
verifying the solution
identifying and correcting errors in a solution
to a quadratic equation
analyzing a problem and then determining
and solving the quadratic
Quadratic Functions and Equations
Page 32 of 50
Lesson Summary

Students will review factoring polynomials of degree 2 (quadratics).
Lesson Plan
Activating Prior Knowledge
Factor polynomials
Lesson
Review factoring and expand to more complex quadratics.
Going Beyond
Factor more complex quadratics (rational coefficients).
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 4.3)
Supporting
Assessment
Create a formative assessment on factoring.
Mathematics 20-1
Quadratic Functions and Equations
Page 33 of 50
Glossary
binomial – A two term polynomial expression
factor – To express an algebraic expression as a product
trinomial – A three term polynomial expression
Other
Mathematics 20-1
Quadratic Functions and Equations
Page 34 of 50
Lesson 4
Solving/Graphing Quadratics by Factoring or Quadratic Formula
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …


Solving an equation means to find the root(s).


Find two examples of real-life situations that
could be represented by a quadratic function.
In what situations is one method of solving a
quadratic equation preferable to another?
Given y = a(x - p)2 + q and the values of a and
q, how can you determine the number of xintercepts on the corresponding graph?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …


the zeros of a function correspond to the xintercepts of a graph







solve a quadratic equation of the form
ax2 + bx + c = 0 by:
determining square roots
factoring
completing the square
applying the quadratic formula
graphing its corresponding function
verifying the solution
identifying and correcting errors in a solution
to a quadratic equation
Lesson Summary



Solve Quadratics by Graphing
Solve Quadratics by Factoring
Introduce the Quadratic Formula to solve Quadratics
Lesson Plan
Hook
Solve a linear equation algebraically and graphically.
Mathematics 20-1
Quadratic Functions and Equations
Page 35 of 50
Learning Goal
Students will solve quadratics by graphing, factoring and the quadratic formula.
Activating Prior Knowledge
Ask students to factor some quadratic functions to check for understanding of the
previous lesson.
Lesson
Discuss the importance and what it means to solve a quadratic equation.
Graphing
 Provide a set of quadratics to be solved using technology.
Algebraic
 Provide an increasing more complex set of quadratics to be solved by factoring.
(integral, rational and exact value roots)
 Provide the Quadratic Formula (extension derive the formula)
-b ± b 2 - 4ac
2a
Discuss the uses of the quadratic formula
x =

Going Beyond
1. Derive the quadratic formula.
2. Introduce/discuss non-real roots.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 4.1, 4.2, 4.4)
Mathematics 20-1
Quadratic Functions and Equations
Page 36 of 50
Supporting
Assessment
Exit Slip
Two quadratics solved algebraically
one factorable
one non-factorable (exact value)
Glossary
exact values – Answers involving non-rounded values, such as whole numbers,
terminating or repeating decimals, fractions or radicals.
roots – Find the solution(s) to an equation
solve – To find the solutions to an equation
x-intercept – The x-coordinate of the intersection point of a graph and the horizontal
axis
zeros – The values of x that make a function equal to zero
Other
Mathematics 20-1
Quadratic Functions and Equations
Page 37 of 50
Lesson 5
Discriminant
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …


Solving an equation means to find the root(s).

Find two examples of real-life situations that
could be represented by a quadratic function.
Given y = a(x - p)2 + q and the values of a and
q, how can you determine the number of xintercepts on the corresponding graph?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the discriminant (b2- 4ac) can be used to
determine the nature of the roots

use the discriminate to determine the nature
and number of roots
Lesson Summary

Students will use the discriminant to determine the number and nature of the
roots.
Lesson Plan
Hook
Discuss 3 quadratics graphs with 3 different natures of the roots.
Lesson Goal
Students will recognize and make the connection between the numbers of roots and
the discriminant.
Activate Prior Knowledge
Mathematics 20-1
Quadratic Functions and Equations
Page 38 of 50
Knowledge of the quadratic formula
Lesson
Solve the initial three introductory graphs using the quadratic formula.
Discover and discuss the connection between the radicand in the quadratic formula
and the nature of the roots.
Identify and discuss the discriminant.
Going Beyond
Looking at the parameters a, b, c in a quadratic equation and without using the
discriminant, determine the nature of the roots.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 4.4)
Supporting
Assessment
Mathematics 20-1
Quadratic Functions and Equations
Page 39 of 50
Glossary
discriminant – (b2 – 4ac) This value tells the nature of the roots
equal real roots – the discriminate equals zero
non-real number – Imaginary number (such as the square root of a negative number)
non-real roots – the discriminate (b2 – 4ac) value is less than zero
unique real roots – the discriminate (b2 – 4ac) value is greater than zero
Other
The following applet includes most of the glossary terms. Understanding the
display will help students remember quadratic terms.
Source: http://www.ronblond.com/MathGlossary/Division04/Quad01/
Mathematics 20-1
Quadratic Functions and Equations
Page 40 of 50
Lesson 6
Completing the Square
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



The basic shape and characteristics of the
graph of a quadratic function.
Solving an equation means to find the root(s).





Find two examples of real-life situations that
could be represented by a quadratic function.
Get as close as you can to the equation of a
parabola with vertex (-2,5).
Given the graph of the parabola shown, find
the equation.
How does this graph describe the revenue of
a school dance?
Given y = a(x - p)2 + q and the values of a and
q, how can you determine the number of xintercepts on the corresponding graph?
How can algebra tiles be used to model
completing the square?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …


that (p, q) is the vertex
what a, p, q means in the context of a
problem


identify/determine the coordinates of the
vertex
change the form of a quadratic equation by
completing the square or expanding
Lesson Summary

Students will be able to change the form of a quadratic equation by completing
the square and verify.
Mathematics 20-1
Quadratic Functions and Equations
Page 41 of 50
Lesson Plan
Advance Planning
Students will be required to cut out the paper version of the algebra tiles in order to
manipulate the tiles.
http://www.themathlab.com/toolbox/algebra%20stuff/algebra%20tiles.htm
1. Note: The teacher will have to modify the tiles if they choose to use them to
work out the example in the Hook on the next page.
Mathematics 20-1
Quadratic Functions and Equations
Page 42 of 50
2. Alternate tiles resource:
http://www.learnalberta.ca/content/t4tes/courses/senior/math10c/u2/m3/m10c_
m3_template.pdf
Hook
Introduce the topic by doing a class example with the newly created algebra tiles.
Refer to website for examples.
http://www.mathsisfun.com/algebra/completing-square.html
Learning Goal
Students will be able to complete the square to change the form of a quadratic
equation. Students need to be able to verify their answer.
Activating Prior Knowledge
Use the algebra tiles to connect to polynomial terms.
Lesson
Get pairs of students to create their own examples, then exchange with another group.
All pairs should attempt to solve the new questions.
In a group discussion, come up with the key ideas involved in completing the square.
Go through the same example together algebraically. Proceed to an example where:
 b is an odd number
 a is not equal to 1.
Allow students time to practice. Check for understanding (orally).
At the end of the class, have the students complete an Exit Slip to check their
understanding.
Mathematics 20-1
Quadratic Functions and Equations
Page 43 of 50
Going Beyond
Students can expand their results to verify their solutions.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 3.3)
Supporting
Assessment
Complete an Exit Slip to check the students’ understanding of the material. This topic
will also be formally evaluated at a later date.
Glossary
completing the square – A method used to convert a quadratic equation from
standard form (y = ax2 +bx + c) to vertex form (y = a(x - p)2 + q).
Other
Mathematics 20-1
Quadratic Functions and Equations
Page 44 of 50
Lesson 7
Determining Quadratic Equations
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
Students will understand …


The basic shape and characteristics of the
graph of a quadratic function.
Solving an equation means to find the root(s).
ESSENTIAL QUESTIONS:
 Get as close as you can to the equation of a
parabola with vertex (-2,5).
 Given the graph of the parabola shown, find
the equation.
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …







that an equation in the form y = x2 is a
quadratic function where the graph is a
parabola
the effects of changing each parameter in the
function y = a(x - p)2 + q
that (p, q) is the vertex
that x = p is the axis of symmetry
that a determines the direction of opening of
the parabola
that the x-intercept is found where y = 0 and
the y-intercept is found where x = 0
the domain is the set of permissible values of
x in the function and range is all the possible
values of y in the function
 write a quadratic function in the form
y = a(x - p)2 + q or y = ax2 + bx + c given a
graph or a set of characteristics of a graph
Lesson Summary

Students will determine equations of quadratic functions, in standard and vertex
form, given graphs and word descriptions.
Mathematics 20-1
Quadratic Functions and Equations
Page 45 of 50
Lesson Plan
Hook
Consider the following video on projectile motion. Ask students how this is different
than other projectile examples they have looked at. You are looking for them to
indicate the projectile is shot out sideways or dropped instead of propelled upward.
http://www.youtube.com/watch?v=qErh402eJgI
Now show them at least one video of a projectile being shot up at an angle. Ask them
what information they need in order to draw the graph of the projectile motion and
what information is required from the graph to determine the equation of the projectile
motion in vertex (y = a(x - p)2 + q) form.
http://www.youtube.com/watch?v=yLUUFelCkF8&feature=related
http://www.youtube.com/watch?v=FBk_q_BSNLM
http://www.youtube.com/watch?v=NNWrW4UVaRY
Lesson Goal
Students will determine equations of quadratic functions, in standard and vertex form,
given graphs and word descriptions.
Activate Prior Knowledge
Review vertex (y = a(x - p)2 + q) form transformations when a, p and q are changed
(Lesson 1).
Lesson
Give students a mixture of graphing and word problems that include:
 a graph.
 a vertex and one point.
 x-intercepts and range.
 a value, one point and range.
 a value, one point and axis of symmetry.
 a congruent graph, with opposite direction and a vertex.
Mathematics 20-1
Quadratic Functions and Equations
Page 46 of 50
Going Beyond
Given any three points on a parabola, determine the quadratic equation.
Resources
Math 20-1 (McGraw-Hill Ryerson: Chapter 3)
Supporting
Assessment
“Exit Slip” or quiz question… May end the class with a final graph/written description
and have students come up with the equation to match.
Groups create an appropriate equation in response to a graph or description created
by another group.
Glossary
congruent graphs - Two graphs are congruent if they have the same shape and size
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-1
Quadratic Functions and Equations
Page 47 of 50
Lesson 8
Applications of Quadratics
STAGE 1
BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to
solve problems in areas like physics, calculus, engineering, architecture, sports and business.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …


quadratic functions and equations relate to
real life situations


Find two examples of real-life situations that
could be represented by a quadratic function.
How does this graph describe the revenue of
a school dance?
In what situations is one method of solving a
quadratic equation preferable to another?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …







that (p, q) is the vertex
that x = p is the axis of symmetry
that a determines the direction of opening of
the parabola
that the x-intercept is found where y = 0 and
the y-intercept is found where x = 0
the domain is the set of permissible values of
x in the function and range is all the possible
values of y in the function
there are many methods available to solve
problems involving quadratics


identify/determine the coordinates of the
vertex
determine an appropriate domain and range
analyze a problem and then determine and
solve the quadratic equation
Lesson Summary

Apply real life situations to quadratics and visa versa.
Mathematics 20-1
Quadratic Functions and Equations
Page 48 of 50
Lesson Plan
Learning Goal

Apply real life situations to quadratics and visa versa.
HOOK
Youtube videos of projectiles (cannonballs, football field goals….)
Mythbusters videos
Activating Prior Knowledge
This is a culmination of the entire unit of study.
Lesson
Students may encounter the following problems:





Sports
o diving, hammer throw, football, shot-put, javelin, discus, basketball,
volleyball, shooting, golf, etc…..
Area
o maximum area of an enclosed shape, given certain amount of material
Numbers
o maximum or minimum product of two unknown numbers, in one variable,
where one number is the variation of the other
Physics
o distance, speed, time, etc….
Revenue
o Find the maximum revenue for selling lemonade, tickets by studying the
effect increase/decrease of price has on the number of sales.
Provide an example of any of the preceding and let the students explore all the
characteristics of a real life quadratic problem. (ie maximum, minimum, vertex,
equation, domain & range). Encourage students discuss the characteristics verbally,
visually (graphs) and symbolically (equations).
The teacher may follow with a directed lesson on more complex examples.
Mathematics 20-1
Quadratic Functions and Equations
Page 49 of 50
Going Beyond
Resources
Math 20-1 (McGraw-Hill: chapters 3 & 4)
Supporting
Assessment
Transfer Task - RAFT
Glossary/Formula
revenue = price x number of items
Other
Mathematics 20-1
Quadratic Functions and Equations
Page 50 of 50