Algebra 1
Quadratic Functions
Code (Kelly will fill this in)
Course: Algebra 1
Unit #5: Quadratic Functions
Overarching Question:
What patterns of change are modeled by quadratic functions as seen in real-world situations,
and the tables, graphs, and functions rules that represent these situations?
Previous Unit:
This Unit:
Exponential Functions
Next Unit:
Quadratic Functions
Solving Quadratic
Equations
Questions to Focus Assessment and Instruction:
Intellectual Processes(Standards for
Mathematical Practice):
1. How do changes in the values of the parameters in a quadratic function
change the behavior of the graph?
2. What is the relationship between the number of real roots and the graph of
a quadratic equation? Why does this relationship exist?
3. How can you translate among the vertex, standard, and factored forms of
quadratic equations? What is useful about each form?
4. What are the characteristics of quadratic change as shown in a graph,
function rule, table, or real-world situation?
Make sense of problems and persevere in
solving them: Build new mathematical
knowledge of quadratic patterns of change
through problem solving
Construct viable arguments and critique the
reasoning of others: Students make
conjectures about how changing the value of
the parameters impacts the graphs and
tables of quadratic equations. They
communicate their reasoning precisely to
others.
Key Concepts:
Patterns of change in quadratic functions
Transformations of quadratic graphs based
on changes of a, b, and c
SCoPE Curriculum
Vertex Form y = a(x - h)2 + k,
Root Form y = a(x - p)(x - q)
Polynomial Form y= ax2 + bx + c)
Multiple representations
Key features of quadratic graphs: vertex,
axis of symmetry, minimum, maximum
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Algebra 1
Quadratic Functions
Code (Kelly will fill this in)
Lesson Abstract
Students use this applet to explore quadratic functions by changing one of the parameters and
observing the impact on the parent function. Multiple parabolas and their equations can be
displayed using a color-coded format, which allows students to make conjectures about the effect
of the changes. The worksheets provided with the applet contain three activities, although the
applet could be used for other explorations. “Challenge A: What About the Coefficient?” asks the
students to explore four given parabolas. Students write equations for the given parabolas in the
form y = ax2, then explore how changing the coefficient a affects the graphs. “Challenge B: Moving
the Parabolas” asks the students to explore vertical and horizontal translations using the Vertex
Form, y = a(x-h)2 + k. “Challenge C: Comparing Forms” asks students to compare transformations
using Polynomial Form, y = ax2+bx+c and Root Form, y=a(x-m)(x-p).
Common Core Standards
Seeing Structure in Expressions (A-SSE)________________________________
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For
example, interpret P(1+r)n as the product of P and a factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as
(x2)2– (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2+y2).
Creating Equations (A-CED)____________________________________________
Create equations that describe numbers or relationships
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.
Interpreting Functions (F-IF)_______________________________________________
Interpret functions that arise in applications in terms of the context
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 verbal
description of the relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries;
end behavior; and periodicity.
Analyze functions using different representations
SCoPE Curriculum
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Algebra 1
Quadratic Functions
Code (Kelly will fill this in)
7. Graph functions expressed 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.
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.
9. Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger maximum.
Building Functions (F-BF)____________________________________________________
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
Build new functions from existing functions
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
Sequence of Lesson Activities
Lesson Title: Weaving a Parabola Web with the Quadratic Transformer
(http://seeingmath.concord.org/Interactive_docs/QT_Activity.pdf)
Selecting and Setting up a Mathematical Task:

By the end of this lesson
what do you want your
students to understand,
know, and be able to do?





SCoPE Curriculum
Students explain the way rates of change are shown in the graph, table,
and equation of quadratic functions.
In the equation f(x) = ax2 , explain precisely what effect changing the value
of a has on the graph and table. Compare this relationship to the rates of
change of linear functions shown in their graphs.
Using the vertex form, y = a(x-h)2 +k, connect the values of h and k with
the location of the vertex. Understand the horizontal or vertical
movements changes in these parameters cause in the graph of the
quadratic function.
Explain mathematically the impact each parameter change in the vertex,
polynomial(standard), and root(factored) forms has on the graph of the
quadratic function and why the change is reasonable.
Compare the information easily accessed in the vertex, polynomial, and
root forms of a quadratic function.
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Algebra 1
Quadratic Functions
Code (Kelly will fill this in)

In what ways does the task
build on student’s previous
knowledge?

Students have experience with identifying rates of change in linear and
exponential functions, as well as y-intercepts of each function type. These
connections and how they impact the graph and table can help students
as they explore similar effects of change in quadratic functions.

What questions will you ask
to help students access
their prior knowledge?

When graphing a linear equation how does changing the value of b in the
equation y = a + bx change the table and graph of the function?
In the exponential equation y = abx , what happens to the graph when b>1,
as b gets larger and larger? Compare this relationship to increasing the
size of b in y = a + bx.
What change in b in the equation y = a + bx causes the direction of the
line to change?
What changes in a linear equation can move the graph up or down? Why
does this make sense?
What changes in table values are characteristic of linear functions?
What changes in table values are characteristic of exponential functions?



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
Launch:

How will you introduce
students to the activity so
as to provide access to all
students while maintaining
the cognitive demands of
the task?

What will be seen and
heard that indicates that the
students understood what
the task is asking them to
do?
 If all students have access to a computer, demonstrate how to use the
Quadratic Transformer applet. As an alternative to a class demonstration
there is a Warm-Up activity provided that introduces the user to the applet.
Students could pair up for the activity and alternate who is interacting with
the computer technology if there are not sufficient resources available. It is
also possible to download to applet at no charge and then use it offline. The
teacher should circulate to make sure all students understand how to use
the applet in order to explore quadratic functions.
 Students will be commenting on the changes in the shapes of the parabolas
and talking about the different forms in which they can display their
equations.
 Multiple graphs should be on the computer screen, so that students can
make conjectures about what is happening with each parameter change.
Supporting Student’s Exploration of the Task:

What questions will be
asked to focus students’
thinking on the key
mathematics ideas?
SCoPE Curriculum
 For Challenge A, the important idea is the effect of changing the value of a
in the equation y = ax2 .
o As the value of a gets larger, starting with a = 1, what happens to the
shape of the parabola? Why does this make sense? How does this
relationship compare to the changes in linear graphs when the value
of b in the equation y = a + bx continues to grow in a similar
manner?
o Describe what happens to the graph of a quadratic function when the
value of a is negative. How would the tables of the equations y = ax 2
and y=-ax2 compare with one another?
o Many students confuse the relationship of graphs such as y = 4x 2
and y= ¼ x2. They often expect the fractional value to be a more
narrow graph because ¼ is smaller than 4. Ask students explicitly to
consider a situation similar to this. Explain the relationship between
y = 4x2 and y = ¼ x2 in terms of the appearance of the graph and
what a table of values would look like. How does this connect to
ideas of rate of change in linear functions?
 For Challenge B, the focus turns to vertical and horizontal translations.
Although any form could be used, the one that works most easily is vertex
form y=a(x-h)2 +k.
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Algebra 1
Quadratic Functions
Code (Kelly will fill this in)

What questions will be
asked to assess student’s
understanding of key
mathematics ideas?
o What happens to the graph of a quadratic function when the value of
k in the vertex form changes? Describe the movement of the
functions as k increases or decreases.
o What happens to the graph of a quadratic function when the value of
h in the vertex form changes? Describe the movement of the
functions as h increases or decreases. Why does the graph move in
the opposite direction you expected? Give a mathematical
explanation for this.
o Where is the point (h,k) located in the graph y=a(x-h)2 +k?
 Challenge C compares what information is easily accessible about the graph
of a quadratic function depending on the form used. The polynomial
(y=ax2+bx+c) and root forms (y = a(x-m)(x-p)) are explored.
o Using the two different equations given in problem #1, what
information was easily found in each equation? Why does the
polynomial form allow you to easily determine the y-intercept? Why
does the root form allow you to easily determine the x-intercepts?

What will you do if a
student finishes the task
almost immediately? How
will you extend the task to
provide additional
challenge?
 Have students explore the polynomial form, in particular using the applet to
explore the effect of changing the value of b in y = ax 2 +bx+c. Ask students
to observe the path of the vertex as the value of b changes and make a
conjecture about why this is true.

What assistance will you
give a student who
becomes quickly frustrated
and requests more
direction and guidance?
 Students that are having problems with these activities may not be as strong
at the visual aspects of the graphs. Use a graphing calculator to examine
tables to look for numeric connections.
 In Challenge B, ask students to consider the coordinates of the x- and yintercepts, and what the location of the zeroes in these coordinates has to
do with the outcome of the equations.
Sharing and Discussing the Task:

What specific questions will
be asked so that all
students will:
o
Make sense of the
mathematical ideas
that you wanted them
to learn?
o
Expand on, debate,
and question the
solutions being
shared?
o
Make connections
between the different
strategies that are
presented?
o
Look for patterns?
 What happens to the parabola y=ax2 when a>0, as a increases? when a<0,
as a decreases?
 Given an equation of a quadratic function, what is another equation that
causes a vertical translation to the graph? The equation could be given in
polynomial, vertex, or root form.
 Given an equation in either vertex, polynomial, or root form, transform the
equation so that it is translated horizontally. Can you explain why your
change works in each form?
 What information does each form of a quadratic function easily provide?
When would you choose to use one form over another?
 Compare how you transform linear, exponential, and quadratic functions
vertically.
o Begin to form
generalizations?
SCoPE Curriculum
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Algebra 1
Quadratic Functions

How will you ensure that,
over time, each student has
the opportunity to share his
or her thinking and
reasoning with peers?
Code (Kelly will fill this in)
 Have students work in pairs to demonstrate one of the ideas of the lesson
and share their thinking with each other.
 Randomly choose students to demonstrate their understanding of the graph
of a quadratic function.
Formative Assessment
Students can use vertex or root form and their understanding of transformations to write equations
for the six graphs provided. For each problem they should provide an explanation for how they
made their decisions.
http://itech.pjc.edu/falzone/worksheets/transform-parabolas-notes.pdf
SCoPE Curriculum
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