ALGEBRA
Module:
9
Module Name:
Quadratic Functions
Recommended Time Frame
# Weeks
Big Idea
Many real-world situations can be modeled using quadratic equations and expressions.
Prior Learning
Students have spent the year working with linear equations. Quadratic functions are main function type taught in Algebra I. Students should be expected to compare and
contrast the three main function types, looking for similarities and differences in all of the representations. In Algebra II, students will expand on this work when studying
polynomial functions of degree  3 as well as rational and radical functions.
Vocabulary- Common Core
Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should
learn to use with increasing precision with this cluster are: quadratic function, parabola, vertex (of a parabola), zeroes (of a function), standard form (of a quadratic),
vertex form (of a quadratic), factoring, completing the square, the quadratic formula.
Essential Questions
How are quadratic functions the same (or different from) linear functions?
How can we sketch a graph of a quadratic function?
What do the different forms for writing quadratic equations reveal about the function?
What are the characteristics of Quadratic Functions?
How does a quadratic equation define and describe a parabola?
How do I recognize a quadratic function from an equation, table and graph?
How do I decide the best method to solve a quadratic equation?
What happens to the graph of y = x when we change it to y = ax + c?
Students will be able to apply and discuss how to
Factor a quadratic expression to reveal the zeros of the function
Complete the square in a quadratic expression to reveal the minimum or maximum value of the function
Create a quadratic equation that describes a given situation
Graph a quadratic function by creating a table of ordered pairs
Identify the y-intercept, zeroes and vertex of a quadratic function and use that to create a rough sketch of the function
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Solve quadratic equations by inspection, factoring, completing the square and the quadratic equation
Find the average rate of change of a quadratic over a specified interval
Informally identify the effects on a quadratic graph when replacing f ( x ) with f ( x )  k , kf ( x ) , f (kx) and f ( x  k )
CCSS-Mathematics
Content Standards
Examples:
A-SSE.3. Choose and produce an
equivalent form of an expression to reveal
and explain properties of the quantity
represented by the expression. ★
a. Factor a quadratic expression to reveal
the zeros of the function it defines.
b. Complete the square in a quadratic
expression to reveal the maximum or
minimum value of the function it defines.
A-SSE.3a Students factor quadratic expressions and find the zeros of the quadratic function it represents. Zeroes are the x values
that yield a y value of 0. They should also explain the meaning of the zeros as they relate to the problem.
A-CED.1. Create equations and
inequalities in one variable and use them
to solve problems. Include equations
arising from linear and quadratic
functions, and simple rational and
exponential functions.
i) Tasks are limited to linear or
exponential equations with integer
exponents.
A-CED.1 From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include onevariable equations that arise from functions by the selection of a particular target y-value.
Ex. If
(3m2  15m) is the income gathered at a rock concert, what values of m would produce an income of 0?
A-SSE.3b Students rewrite a quadratic expression in the form
k), and explain its meaning in context.
( x  h)2  k , with a = 1, to identify the vertex of the parabola (h,
Ex. Given a context modeled by the quadratic expression x  24 x  55 , find the vertex and interpret its meaning in this context.
2
Ex. A pool can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the pool is full, it can be drained by pipe C in 4
hours. If the pool is initially empty and all three pipes are open, write an equation to find how long it takes to fill the pool.
Ex. A cardboard box company has been contracted to manufacture open-top rectangular storage boxes for small hardware parts.
The company has 30 cm x 16 cm cardboard sheets. They plan to cut a square from each corner of the sheet and bend up the sides
to form the box. If the company wants to make boxes with the largest possible volume, what should be the dimensions of the
square to be cut out? What are the dimensions of the box? What is the maximum volum
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A-CED.2. Create equations in two or
more variables to represent relationships
between quantities; graph equations on
coordinate axes with labels and scales.
A-CED.2 Given a contextual situation, write equations in two variables that represent the relationship that exists between the
quantities. Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations
arising from the functions they have studied.
Ex. The height of a ball t seconds after it is kicked vertically depends upon the initial height and velocity of the ball and on the
downward pull of gravity. Suppose the ball leaves the kicker’s foot at an initial height of 0.7 m with initial upward velocity of
22m/sec. Write an algebraic equation relating flight time t in seconds and height h in meters for this punt.
Ex. The intensity of light radiating from a point source varies inversely as the square of the distance from the source. Write an
equation to model the relationship between these quantities given a fixed energy output.
A-APR.3. Identify zeros of polynomials
when suitable factorizations are available,
and use the zeros to construct a rough
graph of the function defined by the
polynomial.
i) Tasks are limited to quadratic and cubic
polynomials in which linear and quadratic
factors are available. For example, find
the zeros of
( x  2)( x2  9) .
A-REI.4. Solve quadratic equations in
one variable.
a) Use the method of completing the
square to transform any quadratic
equation in x into an equation of the
form (x – p)2 = q that has the same
A-APR.3 Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph.
Ex. Given the function
y  2 x 2  6 x  3 , list the zeros of the function and sketch the graph.
Ex. Sketch the graph of the function f ( x)  ( x  5) . What is the multiplicity of the zeros of this function? How does the
multiplicity relate to the graph of the function?
2
Ex. For a certain polynomial function, 𝑥 = 3 is a zero with multiplicity two, 𝑥 = 1 is a zero with multiplicity three, and 𝑥 = −3 is a
zero with multiplicity one. Write a possible equation for this function and sketch its graph.
A-REI.4a Transform a quadratic equation written in standard form to an equation in vertex form,
the square.
( x  p)2  q , by completing
Ex. Write the quadratic equation, y= 2 x  16 x  20 in vertex form. What is the vertex of the graph of the equation?
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ALGEBRA
solutions. Derive the quadratic
formula from this form.
A-REI.4a Derive the quadratic formula by completing the square on the standard form of a quadratic equation.
Ex. Solve
y  ax 2  bx  c for x. What do you notice about your solution?
PARCC Assessment Limitations for Algebra I
i) Tasks do not require students to write solutions for quadratic equations that have roots with nonzero imaginary parts. However,
tasks can require the student to recognize cases in which a quadratic equation has no real solutions. Note, solving a quadratic
equation by factoring relies on the connection between zeros and factors of polynomials (cluster A-APR.B). Cluster A-APR.B is
formally assessed in A2.
b) Solve quadratic equations by
inspection (e.g., for x2 = 49), taking
square roots, completing the square,
the quadratic formula and factoring,
as appropriate to the initial form of
the equation. Recognize when the
quadratic formula gives complex
solutions and write them as a ± bi for
real numbers a and b.
A-REI.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the
square. Add context or analysis
Ex. Find the solution to the following quadratic equations:
a.
b.
c.
x 2  7 x  18  0
x 2  81
x 2  10 x  5  0
A-REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula always produces solutions. Write the
solutions in the form a ± bi , where a and b are real numbers.
Students should understand that the solutions are always complex numbers of the form
a  bi . Real solutions are produced when
b = 0, and pure imaginary solutions are found when a = 0. The value of the discriminant,
what type of solutions the quadratic equation has.
Ex. Ryan used the quadratic formula to solve an equation and x 
b2  4ac , determines how many and
8  (8)2  4(1)(2)
was his result.
2(1)
a) Write the quadratic equation Ryan started with.
b) Simplify the expression to find the solutions.
c) What are the x-intercepts of the graph of this quadratic function?
F-IF.4. For a function that models a
relationship between two quantities,
interpret key features of graphs and tables
in terms of the quantities, and sketch
graphs showing key features given a
F-IF.4 This standard should be revisited with every function your class is studying.
Students should be able to move fluidly between graphs, tables, words, and symbols and understand the connections between the
different representations. For example, when given a table and graph of a function that models a real-life situation, explain how the
table relates to the graph and vise versa. Also explain the meaning of the characteristics of the graph and table in the context of the
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verbal description of the relationship. ★
problem as follows:
At the course one level, the focus is on linear, exponentials, and quadratics



Linear – x/y-intercepts and slope as increasing/decreasing at a constant rate.
Exponential- y-intercept and increasing at an increasing rate or decreasing at a decreasing rate.
Quadratics – x-intercepts/zeroes, y-intercepts, vertex, intervals of increase/decrease, the effects of the coefficient of x 2 on the
concavity of the graph, symmetry of a parabola.
(Level I)
Ex. Below is a table that represents the relationship between daily income, I, for an amusement park and the number of paying
visitors in thousands, n.
a)
b)
c)
d)
e)
n
I
0
0
1
5
2
8
3
9
4
8
5
5
6
0
What are the x-intercepts and y-intercepts and explain them in the context of the problem?
Identify any maximums or minimums and explain their meaning in the context of the problem.
What pattern of change do these ordered pairs develop? Explain.
Is the pattern and/or graph of the data symmetrical? How do you know?
Describe the intervals of increase and decrease and explain them in the context of the problem.
PARCC Assessment Limitations for Algebra I
i) Tasks have a real-world context.
ii) Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined
functions (including step functions and absolute value functions), and exponential functions with domains in the integers.
Compare note (ii) with standard F-IF.7.
The function types listed here are the same as those listed in the Algebra I column for standards F-IF.6 and F-IF.9.
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F-IF.6. Calculate and interpret the
average rate of change of a function
(presented symbolically or as a table)
over a specified interval. Estimate the rate
of change from a graph.★
F-IF.6 Students should be able to describe patterns of changes from tables and/or graphs of linear and exponential
functions. Sample vocabulary may include, “increasing/decreasing at a constant rate” or “increasing or decreasing at an increasing
or decreasing rate.” Students should be comfortable in their understanding of rates of change to apply their knowledge linear and
non-linear graphical display.
Ex. Kierra is a saleswoman whose pay depends on the number of diamonds she sells. She earns a base salary plus a commission on
each sale. Using the table below, determine the rate of change in earnings as sales increase. What part of Kierra’s pay does this
represent?
Number of Diamonds
Weekly Earnings
2
4
6
8
600
960
1320
1680
Ex. The graph below shows the population of Leicester from 1950 to 1990. Use the graph to answer the following questions.
a) What was the average rate of change from 1975 to 1985?
b) What does this rate of change mean in the context of the problem?
c) Note- this could be used as a standalone question without parts a. and b. If the population increased by the same average rate
from the year 1990 to 2005 as it did from 1975 to 1985, approximately what is the expected population in the year 2005?
Justify your answer.
PARCC Assessment Limitations for Algebra I
i) Tasks have a real-world context.
ii) Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined
functions (including step functions and absolute value functions), and exponential functions with domains in the integers.
The function types listed here are the same as those listed in the Algebra I column for standards F-IF.4 and F-IF.9.
F-IF.7. Graph functions expressed
F-IF.7 Part a., b., and c. are learned by students sequentially in Courses I – III. Part e. is carried through Courses I – III with a
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ALGEBRA
symbolically and show key features of the
graph, by hand in simple cases and using
technology for more complicated cases.★
a) Graph linear and quadratic functions
and show intercepts, maxima, and
minima.
focus on exponential in Course I and moving towards logarithms in Courses II and III. Part d. is introduced in course III with
further development in a fourth course option.
This standard should be seen as related to F-IF.4 with the key difference being students can create graphs, by hand and using
technology, from the symbolic function in this standard.
Ex. The all-star kicker kicks a field goal for the team and the path of the ball is modeled by f (t )  4.9t  20t . Find the
realistic maximum and minimum values for the path of the ball and describe what each means in the context of this problem.
2
F-IF.8. Write a function defined by an
expression in different but equivalent
forms to reveal and explain different
properties of the function.
a. Use the process of factoring and
completing the square in a quadratic
function to show zeros, extreme values,
and symmetry of the graph, and interpret
these in terms of a context
F-IF.8a This standard is developed throughout Courses I - III. In Course I and II, students will factor quadratic functions and in
Course III, students will complete the square.
Students should manipulate a quadratic function to identify its different forms (standard, factored, and vertex) so that they can
show and explain special properties of the function such as; zeros, extreme values, and symmetry. Students should be able to
distinguish when a particular form is more revealing of special properties given the context of the situation.
An exemplar lesson plan of F-IF.8 and F-IF.9 is available from the Shell Center for Mathematics Education titled L20: Forming
Quadratics at http://map.mathshell.org/materials/lessons.php?taskid=224
Ex. In the cartoon, Coyote was chasing the Road Runner, seeing no easy escape, Road Runner jumped off a cliff towering above
the roaring river below. Molly mathematician was observing the chase and obtained a digital picture of this fall. Using her
mathematical knowledge, Molly modeled the Road Runner’s fall using several quadratic functions:
h(t )  16t 2  32t  48
h(t )  16(t  1)(t  3)
h(t )  16(t  1)2  64
a) Does Molly have three unique equations that model the same situation? Explain.
b) Explain what it means for expressions or equations to be mathematically equivalent?
c) Which of the equivalent equations would be most helpful in answering each of these questions? Explain.
i.
What is the maximum height the Road Runner reaches and when will it occur?
ii.
When would the Road Runner splash into the river?
iii.
At what height was the Road Runner when he jumped off the cliff?
F-BF.1. Write a function that describes a
relationship between two quantities.★
F-BF.1a Write equations for functions by developing the NOW-NEXT form or use the context in which the problem is given to
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a) Determine an explicit expression, a
recursive process, or steps for calculation
from a context.
b) Combine standard function types using
arithmetic operations.
find the equation.
F-BF.1b Use addition, subtraction, multiplication, and division to combine functions to create other functions. When used in
context, interpret the effect the combination of the functions had on the given situation.
PARCC Assessment Limitations for Algebra I
i) Tasks have a real-world context.
ii) Tasks are limited to linear functions, quadratic functions, and exponential functions with domains in the integers.
F-BF.3. Identify the effect on the graph of
replacing f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k (both
positive and negative); find the value of k
given the graphs. Experiment with cases
and illustrate an explanation of the effects
on the graph using technology. Include
recognizing even and odd functions from
their graphs and algebraic expressions for
them.
F-BF.3 Identify the effect transformations have on functions in multiple modalities or ways. Student should be fluent with
representations of functions as equations, tables, graphs, and descriptions. They should also understand how each representation of
a function is related to the others. For example, the equation of the line is related to it’s graph, table, and when in context, the
problem being solved.
Ex. Fill in all missing components to the below table.
Description
of Change
Original
Function
Output
x
f ( x)
-3
9
-2
4
-1
1
0
0
1
1
2
4
3
9
Multiply the
original function
output by 5
f ( x)  5
5
9
a)
Graph and label each of the function outputs with the corresponding x-values on the same set of axis in three different
colors.
b) Explain the relationship that exists between the original function and the transformed functions.
PARCC Assessment Limitations for Algebra I
i.
Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k (both
positive and negative) is limited to linear and quadratic functions.
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ii.
iii.
Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear
functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step
functions and absolute value functions), and exponential functions with domains in the integers.
Tasks do not involve recognizing even and odd functions.
The function types listed in note (ii) are the same as those listed in the Algebra I column for standards F-IF.4, F-IF.6, and F-IF.9.
Suggested Resources from The Key Elements to Algebra Success - KEAS
for Building the Conceptual Understanding of this Module.
Lesson 33:
Lesson 34:
Lesson 35:
Lesson 36:
Lesson 37:
Factoring Polynomials
Factoring Trinomials
Factoring Special Cases
Graphing Quadratic Equations
Graphing Quadratic – Word Problems
Additional Resources for KEAS
Lesson 33: Factoring Polynomials
Additional Activities: Quiz - T810, Activity - Scavenger Hunt T1267
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.keasmath.com/lesson%20pages/Lesson%2033%20Factoring%20Polynomials.html?utm_source=KEASELetter33&utm_medium=email&utm_campaign=KEASL
esson33
Lesson 34: Factoring Trinomials
Additional Activities: Quiz - T847, Activity - Scavenger Hunt T1269
Foldable: “Factoring” (box method foldable)
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.keasmath.com/lesson%20pages/Lesson%2034%20Factoring%20Trinomials.html?utm_source=KEASELetter34&utm_medium=email&utm_campaign=KEASLes
son34
Lesson 35: Factoring Special Cases
Additional Activities: Quiz - T871, Activity – Chain Reaction T1271
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.keasmath.com/lesson%20pages/Lesson%2035%20Factoring%20Special%20Cases.html?utm_source=KEASELetter35&utm_medium=email&utm_campaign=K
EASLesson35
Lesson 36: Graphing Quadratic Equations
Additional Activities: Quiz - T904, Activity - Scavenger Hunt T1273
Foldable: “Quadratics”(4 page booklet foldable)
To access the online teacher lesson notes, a video clip, and the student homework go to:
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ALGEBRA
http://ntnmath.keasmath.com/lesson%20pages/Lesson%2036%20Graphing%20Quadratic%20Functions.html?utm_source=KEASELetter36&utm_medium=email&utm_camp
aign=KEASLesson36
Lesson 37: Graphing Quadratic – Word Problems
Additional Activities: Quiz - T931, Activity - Chain Reaction T1275
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.keasmath.com/lesson%20pages/Lesson%2037%20Quadratic%20Functions%20Word%20Problems.html?utm_source=KEASELetter37&utm_medium=email&utm_campaign=KEASLesson37
Other Resources
Additional Manipulatives/ Tools to Enhance the Module
Texas Instruments (
http://education.ti.com/calculators/timath/)
Applications of Parabolas(TI-84+): In this activity, students will look for both number patterns and visual shapes that go along with quadratic relationships. Two
applications are introduced after some basic patterns in the first two problems.
http://education.ti.com/xchange/US/Math/AlgebraI/12236/Alg1_AppsParabolas_TI84.pdf
Exploring the Vertex Form of the Quadratic Function(TI-84+): Students explore the vertex form of the parabola and discover how the vertex, direction, and width of
the parabola can be determined by studying the parameters. They predict the location of the vertex of a parabola expressed in vertex form.
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5560
NCTM Illuminations(
http://illuminations.nctm.org/)
Egg Launch Contest: Students will represent quadratic functions as a table, with a graph, and with an equation. They will compare data and move between
representations.
http://illuminations.nctm.org/LessonDetail.aspx?id=L738
Building Connections: This lesson focuses on having students make connections among different classes of polynomial functions by exploring the graphs of the
functions. The questions in the activity sheets allow students to make connections between the x-intercepts of the graph of a polynomial and the polynomial's
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factors. This activity is designed for students who already have a strong understanding of linear functions, some knowledge of quadratic functions, and what is meant
by a polynomial function.
http://illuminations.nctm.org/LessonDetail.aspx?id=L282
Hanging Chains: Both ends of a small chain will be attached to a board with a grid on it to (roughly) form a parabola. Students will choose three points along the curve
and use them to identify an equation. Repeating the process, students will discover how the equation changes when the chain is shifted.
http://illuminations.nctm.org/LessonDetail.aspx?id=L628
Roller Coasting Through Functions: In this lesson, students determine the time it takes for a roller coaster to reach the bottom of its tallest drop. They use tables and
graphs to analyze the falls of different roller coasters. Students conclude the study by creating their own roller coaster and providing an analysis of its fall.
http://illuminations.nctm.org/LessonDetail.aspx?id=L839
Performance Tasks
Zeros and factorization of a general polynomial.
http://illustrativemathematics.org/illustrations/788
Zeros and factorization of a quadratic polynomial Il.
http://illustrativemathematics.org/illustrations/787
Zeros and factorization of a non polynomial function.
http://illustrativemathematics.org/illustrations/796
Zeros and factorization of a quadratic polynomial II.
http://illustrativemathematics.org/illustrations/789
Misconceptions
Students may believe that quadratic functions are abstract and exist only “in a math book,” without seeing the usefulness of these functions as modeling real-world phenomena.
Additionally, they believe that the labels and scales on a graph are not important and can be assumed by a reader, and that it is always necessary to use the entire graph of a
function when solving a problem that uses that function as its model.
Some students may believe that factoring and completing the square are isolated techniques within a unit of quadratic equations. Teachers should help students to see the value
of these skills in the context of solving higher degree equations and examining different families of functions.
Students may think that the minimum (the vertex) of the graph of y = (x + 5)2 is shifted to the right of the minimum (the vertex) of the graph y = x2 due to the addition sign.
Students should explore examples both analytically and graphically to overcome this misconception. Some students may believe that the minimum of the graph of a quadratic
function always occur at the y-intercept.
Some students may think that rewriting equations into various forms (taking square roots, completing the square, using quadratic formula and factoring) are isolated techniques
within a unit of quadratic equations. Teachers should help students see the value of these skills in the context of solving higher degree equations and examining different
families of functions.
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Works Referenced in the Development of Modules
Common Core State Standards Initiative http://www.corestandards.org/
North Carolina Mathematics Wiki http://maccss.ncdpi.wikispaces.net/
Illustrative Mathematics Project http://illustrativemathematics.org
Mathematics Assessment Project http://map.mathshell.org.uk/materials/index.php
Assessment Questions to Help Guide Instruction
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Quadratic Functions Mathematical and Real
World Examples
Quadratic Tables
Quadratic Graphs
Quadratic Equations
Symmetry
Symmetry
Vertex Form
Common Second
Differences
Minimum or Maximum
Values
Standard Form
Factored Form
Factoring
Completing the Square
Quadratic Formula
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