YOUNGSTOWN CITY SCHOOLS
MATH: ALGEBRA II
UNIT 1E: POLYNOMIAL, RATIONAL, AND RADICAL RELATIONSHIPS-PART V (3 WEEKS) 2013-2014
Synopsis: Students learn the concept of graphing polynomial functions. They first graph the functions roughly and then
extend this to forming a more accurate graph, showing zeros, discussing end behavior and discussing continuity.
STANDARDS
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.
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.★
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and
showing end behavior.
MATH PRACTICES
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1
L.2
L.7
L.8
L.9
Learn to read mathematical text - - problems and explanations
Communicate using correct mathematical terminology
Research mathematics topics or related problems
Read appropriate text, providing explanations for mathematics concepts, reasoning or procedures
Apply details of mathematical reading / use information found in texts to support reasoning, and develop a “works cited
document” for research done to solve a problem.
MOTIVATION
TEACHER NOTES
1. Teacher mentions the following problem as one that students will learn to work in this unit: the
problem deals with fitting speakers into the trunk of a car. The volume of the trunk of the car
that can contain two 12 in. speakers can be expressed as x3 –12x2 + 47x – 50. Find the value
for x if the volume of the trunk is 10 cu. ft. Students need to understand that they will learn the
method for solving this type equation by graphing. At this time the teacher can discuss other
problems involving volume. (F.IF.7c, MP.1, MP.2, MP.4, MP.5, L.2)
2. Preview expectations for end of unit.
3. Have students set both personal and academic goals for this Unit.
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 1
TEACHING-LEARNING
TEACHER NOTES
Vocabulary: give students the following terms and have them research the terms and develop a definition
in their own words with examples.
Zeros of a function
rational root theorem
end behavior
odd functions
even functions
constant term
leading coefficient
x-y chart
rational roots
synthetic division
continuity
polynomial functions
completing the square
factorization
(L.1; L.7; L.8; L.9)
If your students do not have a graphing calculator, they can use the following on line calculators:
http://go.hrw.com/math/midma/gradecontent/manipulatives/GraphCalc/graphCalc.html
http://www.meta-calculator.com/online/
1. Quick review of skills: Students need to know how to plot points, make an x and y chart, and graph.
They also need to review synthetic division and/or long division, and factor different types of problems
using these skills.
2. Recall: Start with a simple example such as f(x) = x2
x 12 and have students try to graph using
any method they know. Teacher then looks at method a student used, then shows students a new
method for graphing polynomial functions: (A.APR.3; F.IF.7c, MP.2, MP.4, MP.7, L.2)
a.) Look at the zeros of f(x) which we find factoring. We get (x
4)(x + 3), so the zeros are 4, 3 which gives us the ordered pairs (0,4) and (0, -3).
b.) Have the students make an x-y chart or a table. When making the chart, pick a point less
than -3, between -3 and 4, and greater than 4.
c.) Plot points and graph. This gives a rough graph of the polynomial function
3. Discuss end behavior of the function in T/L/ step #2 (function goes to infinity on both ends). Extend
this to odd and even functions. Examples that you could use are f(x)= x4 + x3 – 4x2 – 4x (positive even),
f(x)=x3 – 4x2 + 5 (positive odd), f(x)= -x4 + 5x2 – 2x – 1 (negative even), f(x)= -3x3 + 20x2 – 36x +16
(negative odd). An even degree polynomial has the same end behavior in both the positive and
negative direction. An odd degree polynomial has opposite end behavior in the positive versus the
negative directions. Refer to page 349 in text book or at the table attached on page 6 of the unit
and/or http://www.youtube.com/watch?v=-LJ5Bt8UwCo&feature=related (F.IF.7c, MP.2, MP.7, MP.8,
L.2)
4. Discuss continuity of polynomial functions. Give students an example of a discontinuous function
such as a step function so they can visualize the difference.
TEACHER GENERATED QUIZ AT THIS TIME
5. The rational root theorem (or rational root test) states a constraint on rational solutions (or
roots) of the polynomial equation
with integer coefficients.
If a0 and an are nonzero, then each rational solution x, when written as a fraction x
=p/q in lowest terms satisfies

p is an integer factor of the constant term a0, and
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 2
TEACHING-LEARNING

TEACHER NOTES
q is an integer factor of the leading coefficient an.
Thus, a list of possible rational roots of the equation can be derived using the formula
The Rational Root Theorem assists choosing some possible zeros (roots) to test for the
x-y chart and gives a cut off on the left and right ends of the graph. Use the example f(x)
= 2x2 + 5x – 3. The leading coefficient is +2 and the constant term is +3. The factors of ±2
are ±1 and ±2 and the factors of ±3 are ±1, ±3. So p/q could be ±1/1, ±1/2, ±3/1 and
±3/2. Therefore, the smallest root could be -3 and the largest could be 3. When making
your x- y chart, values need to be between -3 and 3.
Teacher uses discretion on the following:
To help remember which one is p and which is q look at 2x2 – x – 1=0 which is (2x+1)(x–
1)=0 so solving this gives x=-1/2 and 1 so -1/2 means p=1 and q=2.
Add more examples of rational root theorem, as needed (A.APR.3, MP.2, MP.7, MP.8, L.2)
6. Have students work on real-life problems using the rational root (zero) theorem. There are excellent
problems on pages 376 and 381 in the textbook.
TEACHER GENERATED QUIZ AT THIS TIME
7. Graph f(x)=2x3–3x2–8x–3 (A.APR.3; F.IF.7c, MP.1, MP.2, MP.4, MP.5, MP.7, L.2)
a.) Use the rational root theorem to find possible rational roots. Remember to simplify. (p/q)
P=±1, ±3, and q=±1, ±2 and p/q=±1, ±3, ±½. This means that our domain for our zeros are
-3 < x < 3.
b.) Teacher choice: Either use synthetic or long division to find one zero, which is -1, so we have
(x+1)(2x2–3x–3)
Once the expression is reduced to a quadratic, you can use any preferred method to get the
answers.
c.) Set factors equal to zero to find the roots. X= -1, -½, 3
d.) Create an x y chart consisting of the zeros, one point before the smallest zero, one point in
between each zero and one point after the last zero (sections).
e.) Plot and graph.
f.) Discuss end behavior
g.) Discuss continuity.
8. Graph examples such as: (A.APR.3; F.IF.7c, MP.1, MP.2, MP.4, MP.5, MP.7, L.2)
a.) f(x)= - x3 + 6x2 – 11x + 6
b.) f(x)=x3 + 2x2 – 5x
c.) f(x)=x3 +11x2 + 35x + 32.
d.) f(x)=x3 – 2x2 – 5x + 6
9. Explore the following functions on worksheet attached on pages 7 - 8. Pay particular attention to
factors of degree greater than one. Students should surmise if a factor is squared, the graph looks
like a parabola at that zero and if it is cubed, it looks like f(x)=x3 at that zero. (A.APR.3; F.IF.7c,
MP.1, MP.2, MP.4, MP.6, MP.7, L.2)
10. Graph the following examples with students by plotting zeros; fill in the x-y chart. Make sure to
discuss end behavior and continuity : (A.APR.3; F.IF.7c, MP.1, MP.2, MP.4, MP.5, MP.7, L.2)
a.) f(x)=(x-3)2 (x+1)
b.) f(x)= (x+2)3 (x–1)
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 3
TEACHING-LEARNING
TEACHER NOTES
c.) Do more examples as needed.
11. Discuss with students the relationship between zeros and the factored forms. The zeros are found
directly by setting the factors equal to zero and solving. (A.APR.3, MP.2, MP.4, MP.7, L.2)
12. Examine with students different ways to denote the same function. Such as, for parabolas:
f(x)=x2 – 4x – 12 or f(x)= (x–6)(x+2). Then complete the square to show third form. (F.IF.7c, MP.2,
MP.4, MP.7, L.2)
a.) The number in front of x2 needs to be one. If not, divide through by the coefficient in front of
x2.
b.) The x2 and x terms need to be on the same side of the equation by themselves with a
constant on the other side. f(x)+12=x2 – 4x
c.) Take half of the number in front of x, square it, and add it to both sides of the equation.
½*4=2, 22=4
f(x) +12+4=x2 – 4x + 4
f(x) +16= x2 – 4x + 4
d.) Factor the perfect square and get f(x) by itself.
f(x)=(x–2)2 – 16 which is the 3rd form of f(x) = x2 – 4x–12
e.) Practice completing the square to find the vertex.
http://www.compuhigh.com/demo/precalculus/precalcdemolesson02.htm
13. Have students graph the two parent functions: f(x) = x2 and f(x) = x3, discuss with students horizontal
shifts, vertical shifts, stretching and shrinking (f(x) + k, f(x+k), f(kx), kf(x)). Have them graph numerous
examples and arrive at conclusions for these transformations. Relate this to the vertex form of a
parabola.
TRADITIONAL ASSESSMENT
1. Paper-pencil test with M-C questions
TEACHER NOTES
TEACHER CLASASROOM ASSESSMENT
1. Teacher Quizzes and worksheets
2. 2- and 4-point questions
TEACHER NOTES
AUTHENTIC ASSESSMENT
1. Students evaluate goals.
TEACHER NOTES
2. You are the manager of a packaging company responsible for manufacturing
rectangular containers, without a top, from a sheet of cardboard. The containers are
formed by cutting out congruent squares at each corner of the cardboard and folding
up the sides to form a box without a top. The length and width of the cardboard are
constant, however the squares can vary in size. (A.APR.3, F.IF.7c, MP.1, MP.2, MP.3,
MP.4, MP.6, L.2, L.5) Rubric attached on next page
a) Choose a length and width for your piece of cardboard. Using a straight-edge,
draw a diagram marking the length and width with your measurements. Draw and
mark the sides of the squares that are to be removed with a variable. Place
dashed lines on the drawing where the base of the container is to be.
b) Express the volume of the box as a polynomial function, V(x).
c) Graph the function showing the zeros.
d) Find the maximum volume for your box.
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 4
e) List and explain the constraints on the values of x for this problem.
AUTHENTIC ASSESSMENT RUBRIC
ELEMENTS OF THE
PROJECT
Drew diagram and marked
dimensions, and placed
dashed lines for the base
0
Did not
attempt
Found volume of box
Did not
attempt
Completed Graph V(x)
Did not
attempt
Determined Maximum
volume
Did not
attempt
Identified Constraints on x
Did not
attempt
1
2
3
Drew diagram, did
not mark dimensions
and did not place
dashed lines on the
drawing for the base
of the container
Several errors in
expressing volume
of box
Several errors in
graphing function
and did not show
zeros
Found the side of the
square but did not
substitute in to find
volume
Listed and explained
only one constraint
Drew diagram and
marked dimensions, but
did not place dashed
lines for base
Drew diagram, marked
dimensions and x;
placed dashed lines for
base
One error in expressing
volume of box
Volume of box
expressed correctly
Graphed correctly, did
not show zeros
Graphed correctly and
showed zeros
Found volume, but with
a computational error
Found volume
correctly
Listed and explained
two constraints
Listed and explained
all three constraints
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 5
T-L #3 Information chart on end-behavior
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 6
T-L #9
Name:
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 7
-(x+2)^3*(x-1)
1/2*x^3*(x-3)
f(x)=(x-4)^3*(x-1)
x^3(x+2)^2
7/01/2013 YCS Algebra II: Unit 1E: Polynomial, Rational, and Radical Relationships Part V 2013-2014 8