# Mr. Germanis' Class Website

## Chapter 3: Polynomial Models

Date
Activity
Assignments and Homework
1/27
Complete the following tasks in order:
1. List everything you think you know about polynomials.
2. Visit the following website
3. Select an intricate drawing Look at the equations on the left panel of the demonstration.
https://www.desmos.com/art

Introduction to polynomial functions.
(Case study of Taylor Polynomials)

Work on Lesson 3.1 (pp. 179-181)

JR: Why must the sum, difference, and product of polynomials always be a polynomial?
pp.182-183 #4, 7

Uses of Polynomials:
Phases of the Moon (USNO)
Pixar and Points

Today's Goal: Student's will understand at least one example of polynomial use and will add, subtract and multiply polynomials together.
1/28
(Block)
Write the general form of quadratic, cubic and quartic polynomials.
Building understanding of Polynomials and their roots:

Building Polynomials Worksheet

Discussion Zeros and Real Roots for higher order polynomials.

JR: Provide an outline for finding polynomials which pass through particular points on the x-axis.
Complete the following activity to extend today's ideas about linear functions to higher order polynomials. Cubic Polynomial Worksheet

Today's Goal:
Students will learn about roots (aka "zeros") of a polynomial function. Students will learn the minimum number of points to determine a unique polynomial.
Reflection of Final Assessment. Use the THIS document as a guide.

Note: To communicate this reflection, you may type, hand write, video tape, audio record, PowerPoint present or use other mediums. You must answer all questions to receive full credit.

JR: Why is it important to reflect on assessments?
Complete today's activity. Print your document to turn in on Monday.
2/2
Pass forward "Building Polynomial Worksheet" and your Assessment Reflection.

What is The Quadratic Formula? When is it most appropriate to use?

Bonus: Use the general quadratic function to derive the quadratic equation by completing the square.
Explore the solution to a cubic function.
The Cubic Formula.

Fit a curve to points and determine roots given a function.

Factoring and finding roots (i.e. Zeros)

JR: Compare and contrast curve modeling and curve fitting.
Create cubic polynomials that have the following solutions without using the regression function on your calculator. An alternate option may be guess and check. Show all steps of the process and prove  (by hand) the given solutions are correct.
a) (-2, 0), (1, 0), (3, 0)
b) (-2, 0), (-1, 0), (0, -4)
c) (0, -3), (-2, 0), (2, 0)
d)  (-3, 0), (-1/2, 0), (2, 0), (3, 0), (0, 3)
Be prepared to show the graphs (of the polynomials) in class when your homework is checked.

Today's Goal: Students will use points on the x-axis and points not on the x-axis to find a polynomial that fits the data.
2/3
Find a polynomial which passes through (0,0), (0, 4), (0, -3).

How might you find the polynomial if you were given points not on a particular axis?
Motivating Question: What happens when we only know other points?

Lesson 3.2 (pp. 185 - 189)

Expanding polynomials.

JR In your own words, detail the important information you need to remember in order to use points to create a polynomial.
Recreate the polynomials from Feb. 2nd using techniques learned in lesson 3.2. Be prepared to show the matrices (and their inverses) used to calculate the polynomials. Expand each polynomial.

Compare with homework from 1/30 to ensure you have similar solutions.

Today's Goal: Students will learn how to find at least one function to fit points not on the x-axis.

2/4
(Block)
Get ready for the quiz!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.
Quiz 3.1 (write an expanded polynomial that has specified real roots and verify that points lie on a polynomial). You may not use a calculator or notes for this quiz.

After Quiz:
Using graphing calculator to find intersections, and zeros.

Continue Exploring how to fit curves.

JR: Compare and contrast the meaning of the zero of an equation, the roots of an equation and the solutions to an equation.
Pg. 190- 191: #1, 3, 5

Today's Goal: Continue exploring how to find polynomials which pass through points and assess student understanding of roots.

2/6
If $f\left(x\right) = x^2+2x-3$
and $g\left(x\right) = 2x+1$, determine algebraically the values of x where f(x) = g(x).

Graph f(x) and g(x) and use the "intersect" function on your calculator to confirm your algebraic solution.
Polynomial Playtime Activity

Use THESE notes as a guide if needed.

JR: A toy ball is picked up from a tall shelf and tossed across a large gymnasium. Recall that a parametric function can model the vertical height of the ball after an elapsed time. Suppose the height equation (in units of feet) is modeled by h(t) where t is the number of seconds after the ball has been tossed.
$h\left(t\right)=-16t^2+37t$
$h\left(t\right) = -16t^2+37t$Some values of t do not make sense in the context of this problem. What interval(s) of t does this equation make sense? Which interval(s) of t does the equation not make sense?

Be prepared to show your notes, which must include:
• Detailed solutions to all examples.
• Define vocabulary words.
Consider making a Foldable or other note-taking tool to help.
2/9
Let f(x) 2x3 + 2x2 - 6x
1. Factor the polynomial into simple terms.
2. Determine several points of the polynomial.
3. Based on last night's homework, predict how the following function will look by sketching a graph.
End Behavior of a Polynomial

JR: Calculate the polynomial that passes through the points (-1, 49), (0, 15), (1, -1), (2, -5). Using only the equation generated, predict the end behavior of the function.
Complete today's activity
Pg. 206: # 1, 7, 11 (Sketch all graphs into your homework notebook).

Today's Goal: Students will practice fitting polynomials to points, learn about end behaviors of polynomials and scale a polynomial to fit points.
Optional Practice Activity
The activity on the left is optional, but provides good practice for using the quadratic formula to find the roots.
Use The Quadratic Formula to obtain the roots of the polynomial in the ET (from 2/9).  Write the polynomial in factored form.

• Choose non-zero integers for  p, q, and r (avoid double-digit integers).  Substitute them into the equation y = (px - q)(x - r).
• Expand and write in standard polynomial form.
• Select numbers to substitute into x such that -3 ≤ x ≤ 3 then compute the corresponding y-value.  Create three ordered pairs.
Optional activity to provide extra practice.
Falling from the Plane Activity

Summative Question: What are the repercussions of multiplying a polynomial by a scalar?
2/10
Find the roots of the following polynomial:
$y=x^2+4x+5$
Simplify your results as much as possible.

Mathematical Duals: Video Read the story on pg. 210.

The history of Integers, Rational and Irrational Numbers.

Complex numbers introduction. Distinguish between imaginary numbers. Khan's Version of today's lesson.

JR: Expand (i.e. "FOIL") [x - (2 + i)] [x - (2 - i)]
Watch this Khan (Video)

Correctly answer at least 5 problems. You must write down the problem, your solution and the correct answer. (Note: You may need to complete many more than the required number to get at least 5 correct).

HOMEWORK PROBLEMS HERE

Today's Goal: Students will learn how mathematical challenges advanced the common understanding of mathematics. Additionally students will be able to use algebra with complex numbers and reduce their function to standard form $a+bi$ where a and b are real numbers.

2/11
(Block)
Get ready for the quiz!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.

Quiz 3.2 (Fit a curve to points, determine the degree of the function created, identify the roots of the function). Note: No materials (including batteries) will be issued or loaned during class time.

When finished with the quiz: Find the roots of $f\left(x\right) = x^2+4$.

Establish the Fundamental Theorem of Algebra.

Pg 216 #3, work though this problem as a class.

JR: Sketch the polynomial y = x5 - 5x4 + 20x2 - 5x + 25 and determine the roots. How many roots belong to this function? Estimate the x-value for the roots that are real (if any exist). How would you find the complex roots?
Pg 216-217 #1, 3, 4

Today's Goal: Students will understand the utility of the Fundamental Theorem of Algebra (FTA). Additionally, students will show that the FTA is true for quadratic polynomials.
2/13
Use the Fundamental Theorem of Algebra to describe the roots of the function y = x2 - 6x + 13.
Plotting complex numbers in rectangular and polar coordinates.

Complex Plane Overview

JR: Explain why the rectangular and polar forms of a complex number represent the same number.
Pg. 219 #7, 10

You know complex numbers can be represented in both rectangular and polar forms. The complex number can be represented both in polar and rectangular form.
• Draw the complex number above.
• Describe the number in both polar and rectangular form.

Explain why = 8 using both polar coordinates and algebra.

Today's Goal: Students will represent complex numbers on the complex number plane using rectangular and polar coordinates.
2/18
(Block)
A point in polar coordinates can express a number in the complex plane.  What complex number is expressed  by the point (1, 45°)?

Practice using the complex plane and calculating distances.

Complex Hexagonal Jigsaw Puzzle

For those interested:
A paper about complex number and springs

JR: Describe the relationship between the product of -3 + 2with it's complex conjugate and the graphical representation of these numbers.
Complete the following steps for each equation below:
• Draw a sketch of the function.
• Use the FTA to rewrite the function.
• If complex, graph the roots on the complex plane.
• Determine the distance between the roots (explain your calculations).
• Convert the roots to polar coordinates.

1) y = 4x2 - 16x + 19
2) y = 2x2 - 2x + 0.5
3) y = -2x2 + 2x - 3

Today's Goal: Students will extend understanding about complex numbers to find distances between complex numbers (including the distance between the conjugates).
2/20
Get ready for the quiz!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people. Quiz 3.3 (Complex Numbers: Determine real and complex roots, plot a complex number on the complex plane, use the Fundamental Theorem of Algebra to rewrite a polynomial into factored form). You may use a singe 3 x 5 note card and your calculator. Note: No materials (including batteries) will be issued or loaned during class time.

JR: Graph y = (x4 - 13x2 + 36) ÷ (x2 - 2x) in the standard window of your calculator.  Explain the appearance of the function and why you believe it behaves this way.
None Assigned :-)

Consider reviewing content from Quiz 3.1 and Quiz 3.2.

Today's Goal: Assess students knowledge of complex numbers.
2/23
Use long division to calculate 384 ÷ 7 as a mixed fraction. Show all of your work to explain your process. Long division with polynomial functions.

JR: Explain how long division and polynomial long division are similar. Provide an example of each or detail the process.
Pg 233 #8 (Use the long division method only)

Also, compute the following quotients using polynomial division.
• (x3 - x2 - 2x + 6) ÷ (x - 2)
• (2x5 - 7x4 - 13) ÷ (4x2 - 6x + 8)

Today's Goal: Students will calculate P1(x)/P2(x) using polynomial long division. Students will also use the Remainder Theorem to express the remainder of a polynomial division.
2/24
(Block)
For the function f(x) = (x + 2) ÷ (x - 1), describe as many of the following features as possible. DO NOT GRAPH!
• x-intercept
• y-intercept
• end behavior of f (x) as x approaches positive infinity.
• end behavior of f (x) as x approaches negative infinity.
• function behavior when the input is close to 1.
"Be Rational!" (Station Learning Activity)

See THESE notes to clarify information from stations.

JR: Complete the JR for each associated station you visited today. Build a rational polynomial that:
1. Has a hole at x = -10
2. Has a horizontal asymptote at y = -5
3. Has a vertical asymptote at x = 3
4. Intersects the x-axis at (3, 0) and the y-axis at (0, -5)
This homework is due on 3/2.
Note: Some problems may cover concepts NOT at your station. During class, we will continue the activity. All problems should be solvable by the due date.

Pg. 229-233 # 1, 2, 3, 4a, 7

Lesson Goals: Students will learn about basic asymptotic behavior that results from polynomial division including domain and range of asymptotic functions. Additionally, students will recognize a polynomial which results in a curve with a hole.
2/25
(SAT Day)
Visit THIS website from the creators of the SAT.

Explore the links to learn about how the SAT is being redesigned.
Sample SAT Questions (Math, Reading, Writing, Essay)

Sample SBA Assessment (Math and English/Language Arts)

JR: What will you do to prepare for upcoming standardized tests?
(Optional) SAT Prep and sample test on Khan Academy.

Last day to complete extra credit.

Today's Goal: Students will learn about the new SAT and the SBA assessments, how they assess knowledge and practice some problems.
2/27
For the function g(x) = (x + 2)(x - 3) ÷ (x - 1), describe as many of the following features as possible. DO NOT GRAPH!
• x-intercept
• y-intercept
• end behavior of f (x) as x approaches positive infinity.
• end behavior of f (x) as x approaches negative infinity.
• function behavior when the input is close to 1.

Continue "Be Rational!" (Station Learning Activity)

JR: Complete the JR for each associated station you visited today. Build a rational polynomial that:
1. Has a hole at x = -10
2. Has a horizontal asymptote at y = -5
3. Has a vertical asymptote at x = 3
4. Intersects the x-axis at (3, 0) and the y-axis at (0, -5)

See homework column from 2/24. Due 3/2.

Lesson Goals: Students will learn about basic asymptotic behavior that results from polynomial division including domain and range of asymptotic functions. Additionally, students will recognize a polynomial which results in a curve with a hole.
3/2
• Station 1: Why is there a hole in the graph?
• Station 2: Review your response to question 7. Does it make sense?
• Station 3: Review your response to question 6. Does it make sense?
• Station 4: How do you find the x-intercept? How do you find the y-intercept?
Debrief/Overview of polynomial division and asymptotic functions. Practice building rational functions.

See THESE notes to review concepts on rational polynomial functions.

For more practice with identifying attribute of rational functions, use THIS link.

JR: Without a calculator, build a function that
• Has x-intercepts at (-2, 0), (-5, 0), (3, 0).
• Modify your function so there are vertical asymptotes at x = 2, x = 1 and x = -4
• Modify again to have a horizontal asymptote at y = 2/3
Chapter 3 Review Problems

Collaborate on THIS Google Doc. Do not delete others' work.
(Optional: For additional practice problems see book review starting on pg. 242)
3/3
Create a rational function that has an x-intercept at x = 1 and x = 5, a vertical asymptote at x = 0.5 and x = 2, horizontal asymptote at y = -2 and passes through the point (0, -10). In-class review: Questions from homework, activities, and quizzes will be answered.  Bring your papers and ask questions!

JR: Which concept(s) are you least confident? What specific steps will you take to become more confident in this/these skills?
Continue Chapter 3 Review Problems to prepare for Test 3.1 and 3.2

Test review with Mr. G - 3:30pm-4:30pm.

Collaborate on THIS Google Doc. Do not delete others' work.
3/4
(Block)
Get ready for the exam!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people. Put yourself in a positive mental state. Test 3.1.
You may use YOUR calculator and a 3 x 5 note card* (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are also allowed. Note: no materials (including batteries) will be issued or loaned during class time.  Expect questions on:
• Verify roots of a polynomial are solutions.
• Determine the roots of a given polynomial.
• Sketch a graph of a polynomial having specific solutions.
• Use a system of equations to derive a polynomial containing specified solutions.
• Determine the real and complex roots of a given equation.
• Rewrite polynomials according to the Fundamental Theorem of Algebra

*Note cards may not be mechanically reproduced (no photo copies, word processing, etc.).

JR: Describe your own progress towards learning about polynomial functions. What is needed next to improve your understanding?

Continue studying for Test 3.2, send questions to Mr. G.
3/6
Get ready for the exam!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people. Put yourself in a positive mental state. Test 3.2 You may use YOUR calculator and a 3 x 5 note card* (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are also allowed. Note: no materials (including batteries) will be issued or loaned during class time.  Expect questions on:
• Create a rational polynomial with specified intercepts and asymptotes.
• Determine the behavior of a functions when approaching x-values.
JR: Suggest some ways Mr. G's teaching of this unit could be improved. Also, please note some teaching strategies you like.
None :-)

Today is the last day to turn in homework for credit.

Test Corrections: Wednesday 3/11 (AM/PM)
3/9

Which of these functions has a vertical asymptote? Explain how you know.
Slant Asymptotes

From Station 2 of the station activity, sketch a graph of each of the functions in the fifth column on page 3.

JR: Describe all of the possible end behaviors of a function. Consider trigonometric, rational functions, polynomial functions, exponential functions, square root function, etc.
Bring Pressure vs. Altitude graph (from 8 September). If you cannot locate this graph, recreate this for class.

(Recommended) Review exponent rules from Algebra.
Test Corrections: Wednesday 3/11 (AM/PM)