SOTA Fall 2015
Name:
Polynomial Project
Objective: Make connections between the degree of a polynomial and its key features, observe
what happens to the key features when you make simple modifications to the polynomial, and
making sense of intersections.
For this project you will have a project buddy but you will be turning in your own work. It is
broken up into 3 parts for 3 weeks. Each week we will have whole class check-ins (formative
grade) and, as part of that, you and your buddy will be presenting a small piece of the project to
the class so that we can take care of any problem areas as we go. The final project will be due
the last day of class: December 16. This final grade will be summative and weigh as much as
an in-class test.
Part A: The degree of a polynomial and writing a factored form function when the zeros are
known.
A-APR.3
Exceeding
Meeting
Approaching
Beginning
Identify zeroes of
polynomials, factor
polynomials, use the
zeroes to graph
polynomial
functions. (These
zeroes can also be
called x-intercepts.)
Meeting plus:
All 4 polynomials are
correctly given in
factored form with
zeros from list 1.
Some connection is
made between the
degree of a
polynomial and the
number of possible
zeros.
At least 3
polynomials are
correctly given in
factored form with
zeros from List 1.
Less than 2
polynomials are
correctly given in
factored form with
zeros from List 1.
F-IF.7
Exceeding
Meeting
Approaching
Beginning
Graph polynomial
functions expressed
symbolically and
show key features of
the graph.
Meeting plus:
Unique work.
Graphs for 4
polynomials are
included with their
labels. Most key
features are labeled
with their
coordinates: zeros,
max and mins.
Scale is accurate
and graph is pretty
easy to read.
Unique work.
Effort is made to
complete 3 graphs
with their labels.
Some key features
are labeled with their
coordinates: zeros,
max and mins.
Less than 2 graphs
are attempted and
include partial
labeling.
A complete
explanation is given
to question 4.
All key features are
labeled with their
coordinates. Scale
is accurate and
chosen for maximum
display.
1. Choose 4 values for zeros (x-intercepts) between -3 and 3, at least one must be
negative. You and your project buddy should have the same 4 values. List them here.
a. _____________, b. ___________, c. _____________ d. ____________
SOTA Fall 2015
2. Degree 3 polynomials or cubics
Name:
PARTNER A
Create TWO cubic polynomials (degree 3) in factored form with the required number of distinct
zeros from the your list in (1). List your polynomials here.
a. 3 zeros: ____________________________________________________
b. 1 zero: _____________________________________________________
On a separate sheet of graph paper (titled “Degree 3 Polynomials”), graph each one in a
different color and provide a key to indicate which is which. Fill the page vertically. Use a
graphing calculator to help you; label turning points (local maximums and minimums) with their
coordinates, and verify the graph only has zeros that come from your list.
3. Degree 4 polynomials or quartics
Create TWO degree 4 polynomials in factored form with the required number of distinct zeros
from your list in (1). List your polynomials here.
a. 4 zeros: ____________________________________________________
b. 2 zeros: ____________________________________________________
On a separate sheet of graph paper (titled “Degree 4 Polynomials), graph each one in a
different color and provide a key to indicate which is which. Fill the page vertically. Use a
graphing calculator to help you; label turning points (local maximums and minimums) with their
coordinates, and verify the graph only has zeros that come from your list.
4. Generalize: You have now seen degree 1 polynomials (lines), degree 2 polynomials
(quadratics), degree 3, and degree 4 polynomials. How does the number of zeros a
polynomial can have depend on the degree? Explain.
(Record answer here.)
SOTA Fall 2015
Part B: Graph Behavior and the Effects of Small Modifications
Name:
F-IF.2, 4
Exceeding
Meeting
Approaching
Beginning
Use function
notation, evaluate
functions for inputs
in their domain.
Interpret key
features of graphs
and tables in terms
of the quantities;
intercepts, intervals
where increasing or
decreasing, end
behavior, relative
max and min.
Meeting plus answer
to #3a is correct and
well reasoned.
Tables are
complete, interval
lists are correct and
complete, end
behavior is
completely
explained.
Tables from 1a and
2a are mostly
complete, answers
to 1b and 2b are
mostly complete and
end behavior is
discussed.
Table from 1a is
complete or tables
from 1a and 2a are
partially completed.
F-BF.3
Exceeding
Meeting
Approaching
Beginning
Identify the effect on
a graph of replacing
f(x) with f(x) + k and
kf(x).
Meeting and answer
to #3b is correct and
well reasoned.
All 4 constants are
found, table columns
are added, and all 4
graphs are added to
the existing graphs
showing the
resulting new
functions. Question
about change in end
behavior is
answered.
Two constants are
found to meet the
requirements for 1c,
d, 2c or d, table
columns are added
and correctly
computed, graphs
are added to the
existing graphs
showing the
resulting new
functions.
Constants k or C are
attempted and the
graph of the new
function is added to
the existing graphs
to show they work or
column is added to
the table to show the
new function values.
or
answers to 1b and
2b are mostly
complete.
Use a separate sheet of paper to answer all of the parts below except for the graphs. These will
be added to graphs you already have made.
1. Cubics.
a. Using your polynomial from Part A, 2 (a) (having 3 zeros) create a table of values
that includes the zeros, the location of the turning points (max and mins), and a
large negative value for x and a large positive value for x. Compute the y values
for each x value.
b. Based on your table and the graph make a list of the intervals on which the
polynomial is increasing in value and another list of the intervals on which the
polynomial is decreasing in value. Describe the end behavior in words or in
pictures; for positive x values far from 0 what does the graph do, and for negative
x values far from 0 what does the graph do?
SOTA Fall 2015
Name:
c. Modification 1: Looking at your graph and your table modify your polynomial p(x)
by adding a constant k to create p(x) + k so that the graph for y = p(x) + k has at
most 1 zero. Add a column to your table for p(x) + k and compute the y values
for the listed x values. Graph this new function on your graphs of cubics from
Part A.
d. Modification 2: Using your original polynomial p(x) modify by multiplying by a
constant C to create Cp(x) so that the graph for y = Cp(x) has the same zeros as
p(x) but has different end behavior. Add a column to your table for Cp(x) and
compute the y values for the listed x values. Graph this new function on your
graphs of cubics from Part A. How did the end behavior change?
2. Degree 4
a. Using your polynomial from Part A, 3 (a) (having 4 zeros) create a table of values
that includes the zeros, the location of the turning points (max and mins), and a
large negative value for x and a large positive value for x. Compute the y values
for each x value.
b. Based on your table and the graph make a list of the intervals on which the
polynomial is increasing in value and another list of the intervals on which the
polynomial is decreasing in value. Describe the end behavior in words or in
pictures; for positive x values far from 0 what does the graph do, and for negative
x values far from 0 what does the graph do?
c. Modification 1: Looking at your graph and your table modify your polynomial p(x)
by adding a constant k to create p(x) + k so that the graph for y = p(x) + k has no
zeros. Add a column to your table for p(x) + k and compute the y values for the
listed x values. Graph this new function on your graphs of degree 4 polynomials.
d. Modification 2: Modify your original polynomial, p(x), by multiplying by a constant
C to create Cp(x) so that the graph for y = Cp(x) has the same zeros as p(x) but
has different end behavior. Add a column to your table for Cp(x) and compute
the y values for the listed x values. Graph this new function on your graphs of
degree 4 polynomials. How did the end behavior change?
3. Generalize:
a. Can an odd degree polynomial have no zeros? Why or why not?
b. Can you always modify an even degree polynomial so that the resulting
polynomial will have no zeros? Why or why not?