Choose a Function A “Comprehensive Polynomials” Project
Assignment Sheet
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Name: ________________________
Due Date: __________
Standards: AII.P.10: Solves everyday problems that can be modeled using polynomials. Apply appropriate graphical, tabular, or symbolic methods to the solution. Use technology when appropriate. 10.W.5: Write a persuasive proposal that o clarifies and defends positions with precise and relevant evidence, that includes facts, expert opinions, and logical reasoning. o anticipate and address the readers concerns.
Essential Questions: What is the best type of function to model a given scenario? How can I model real life scenarios with various types of polynomial functions? Goal: Use functions to describe and model given scenarios. Role: You are mathematics consultant. As an expert of functions in Algebra II, you will be helping various people in different scenarios to solve their dilemmas by coming up with clever ways to model their situations with equations. You must write a letter to your math teacher to justify your solutions for each scenario. Audience: Your Algebra II teacher Situation: Everyday, people Product, Performance, and Purpose: 1. Read through each given scenario 2. For each scenario, you must come up with the best type of equation to model it 3. Using your toolbox of knowledge from Algebra II, come up with a way to model each scenario with an equation (draw a graph, use Microsoft excel and regression, etc…) 4. Write a letter to your math teacher for each scenario with a justification as to why you selected the type of equation and how well it will model the situation. Use at least 4 words from the word bank in each letter. Use the template to write the letter.
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WORD BANK: Below is a list of words that you should use to guide you in writing the letters to justify why you chose a particular function for a given scenario. You must use at least four of these words for each letter and try to use as many as possible. For each word below, give either a definition or example of it:
Word Definition or Example Word Definition or Example
Linear Slope Y-Intercept Maximum Minimum Quadratic Vertex Axis of Symmetry Roots
Parabola Polynomial Quadratic Cubic Degree Exponential Initial Value Decay Growth
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SCENARIO: BUYING A HONDA CIVIC
Buying a new car is an exciting experience! The stress of getting a loan, dealing with car sales people, figuring out how to pay for it, and choosing the right model and car can be a complicated process. Many people consider how well a car will resale after it has been used. This is called “resale value.” Different cars have different resale values. Recently, the Honda Civic has been rated as one of the cars with the top resale value.
Your job is to project the value of a Honda Civic over time. Let’s say that the retail price of the car is $15,000 plus $20*I for the accessories you want to buy with it. What is the total retail price of the Honda Civic with accessories? $ _____________
Depreciation is the decline in a car's value over the course of its useful life. The older the car is, the more it depreciates. Cars depreciate at different rates that depend on many factors including: consumer demand, the reputation of the model, condition of the car, etc…
According to some research you have done, you have found out that the Honda Civic loses about 11% of it’s value each year. Fill in the value of the car for the first 5 years according to your research.
Age of Car 0 1 2 3 4 5
Value Lost -
Current Value
You want to try your best to take care of your car and keep it for at least 10 years before trying to sell it. Find an equation to model the value of your car over time. Use that equation to calculate the value of your car after 10 years. Equation: _____________________ Value after 10 years: ____________
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SCENARIO: THE MISSING FLIGHT DATA Mr. Schmitz’s Physics class had a paper airplane throwing contest. The winning paper airplane’s trajectory was recorded, but somehow the team lost all but 4 data points. They want to try to recover the entire trajectory of the paper airplane to determine how far it flew and it’s maximum height. Can you help them figure out a way to recreate their data? (HINT: use a regression model with Microsoft Excel to help find the “perfect” model)
x = 20 feet y = (320 – I) feet x = 100 feet y = (560 – I) feet
x = 40 feet y = (500 – I) feet x = 130 feet y = (320 – I) feet
Given the 4 data points above, find an equation that will model the scenario and calculate the distance the airplane traveled, the maximum height, and the missing data points below: x (distance) 0 20 40 60 80 100 120 140 y (height) Equation: ___________________ Roots: ( , ) ( , )
Distance Traveled: ___________ Maximum Height: ___________
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SCENARIO: ANNE’S BUDGET Anne just got a new job as a veterinarian making $43,200 per year. She is trying to save up for a new car and has decided to start a budget to help her keep track of her money. She starts off with $350 in her savings account and writes down her monthly expenses as below:
Expense Rent Cell Phone Groceries Cable TV
Amount (per month) $900 $55 $250 $80
She decides that her budget will be as below. Fill in the missing portions: DESCRIPTION Total Income per month: Total Expenses per month: Savings per month for her car: Extra money per month: AMOUNT $ _________ $ _________ - $20(I) = $ _________ $ _________
Now, she wants to figure out a way to model how much money she can save per month with her current budget. She also wants to figure out a way to model how much money she can save per month if she saves 50% of her extra money per month in addition to how much she has budgeted for her car savings.
Equation #1: ___________________ Equation #2: ___________________
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SCENARIO: BLOWING UP A BALLOON
The Prom committee has tasked you with the job of figuring out how many helium tanks to buy to fill up 10(I) balloons for the prom. The radius of each balloon should be about 50cm so that each balloon will be large and even in size. Each helium tank holds about 50,000,000cm3 of helium. Radius
In order to determine the number of helium tanks to buy, you must do the following in order: 1. Find out the volume of one of your balloons when it is 50cm in radius 2. Calculate the total volume for 10(I) balloons 3. Calculate how many helium tanks to buy depending on the total volume of all the balloons. Below is a table showing the volume of one of your balloons (in cm3) as a function of it’s radius (in cm): Radius (cm) 5 10 15 20 25 30 35 40 Volume (cm3) 533.3 + I = _______ 4,196.7 + I = _______ 14,140 + I = _______ 33,503.3 + I = _______ 65,426.7 + I = _______ 113,050 + I = _______ 179,513.3 + I = _______ 523,721.2 + I = _______
Step one: You must come up with a model for the data listed in the table above (either linear, quadratic, cubic, or exponential). Enter the data into Microsoft excel and find the model that will give you a PERFECT model. Type of Equation: _____________ Equation: _____________________
Use the model to calculate the volume of one balloon that has a radius of 50cm: Volume of one balloon that is 50cm: ______________ cm3
Step two: Calculate the total volume for 10(I) balloons: The volume of 10(I) balloons is: ______________ cm3
Step three: Calculate approximately how many helium tanks you will need to buy: # of helium tanks = Total volume of 10(I) balloons = Volume of one helium tank
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7 On separate sheets of paper, compile a letter for each scenario with attachments based on the format shown below. Outline for Letter to your Math teacher Introduction Paragraph □ explain the purpose of the letter □ describe which function you chose and the scenario Body Paragraph □ explain why you chose the function □ give supporting evidence of why this function is the best Results & Reflections Paragraph □ reflect on what you learned
(Date) Dear Mr./Mrs. (Math teacher’s name), Introduction Paragraph: I am writing you this letter to describe how I was able to model the scenario ____________ by using a ___________ function. The scenario asked me to…
Body Paragraphs: I chose the ___________ function to model my scenario because… You can see that this was a good model for the scenario because… I have attached some supporting evidence of how I determined the best model on the following page (attach any graphs, calculations, etc… to justify your answer).
Conclusion Paragaph: By analyzing this scenario, I learned… Sincerely, (Your Name)
List of Words used from the Word Bank in the Letter (minimum is 4): _______________ _______________ _______________ ______________
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Choose a Function Rubric
Standards: AII.P.10: Solve everyday problems that can be modeled using polynomials. Apply appropriate graphical, tabular, or symbolic methods to the solution. Use technology when appropriate. 10.W.5: Write a persuasive proposal that clarifies and defends positions with precise and relevant evidence; includes facts, expert opinions, and logical reasoning; anticipates and addresses the readers concerns. Exceeds Expectations Vocabulary complete with thorough Word Bank is: definitions or examples Choose your Function Correct function chosen in all 4 cases for each scenario: Calculations Calculations shown for calculations are clear and each scenario: concise for all scenarios Supporting Evidence correct in 4 cases; graphs Graphs for each scenario: given titles and axes labeled Letters Grammar check: all 4 letters are complete with no grammar errors Form of letters: is typed 4 words have been appropriately used in each letter and identified Meets Expectations complete and clear Almost Meets Expectations partially complete Below Expectations incomplete
in 3 cases minor mistakes in calculations for the scenarios correct in 4 cases all 4 letters are complete, but have a few grammar errors is neatly written 4 words have been used in each letter, but not completely appropriately
in 2 cases calculations are only partially correct or have major mistakes correct in 3 cases has many grammar errors or is missing a major point is legible 4 words are used completely inappropriately
in 1 or no cases no calculations are present correct in 2 cases or incomplete is very hard to understand or incomplete is not legible 4 words have not been used in each letter
Word Bank:
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