MATH 30-2

POLYNOMIALS - Module Four

Module 4 - Assignment Booklet

Student: _____________________________ Date Submitted:

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Lesson 1: Polynomial Functions and Their Graphs1. For each of the following polynomial functions, describe the indicated characteristics:

a.

b.

Degree _______________

number of x-intercepts and coordinates of the x-intercepts

y-intercept __________________________________

end behaviour ________________________________

domain _____________________________________

range ______________________________________

number of turning points _______________________

sign of the leading coefficient ___________________

value of the constant term ______________________

Degree _______________

number of x-intercepts and coordinates of the x-intercepts

y-intercept __________________________________

end behaviour ________________________________

domain _____________________________________

range ______________________________________

number of turning points _______________________

sign of the leading coefficient ___________________

value of the constant term ______________________

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c.

2. Write a polynomial function that satisfies each set of characteristics.

a. extending from quadrant II to IV, two turning points, and one x-intercept

b. extending from quadrant II to I and one x-intercept

c. extending from quadrant III to I, degree 1, and a y-intercept of –6

Degree _______________

number of x-intercepts and coordinates of the x-intercepts

y-intercept __________________________________

end behaviour ________________________________

domain _____________________________________

range ______________________________________

number of turning points _______________________

sign of the leading coefficient ___________________

value of the constant term ______________________

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3. The average retail price of a pair of designer jeans in Canada, from 1980 to 2011, can be modelled by the polynomial function P(x) 0.0045x3 0.188x2 4.331x 18.65.

a. Describe the characteristics of the graph of the polynomial function. Include the following:

degree and type of graph end behaviour possible number of turning points

b. Explain what the constant term means in the context of the problem.

c. What is the predicted value of jeans in 2012 using this function?

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4. When designing a road, different types of curves are used. Two common types of curves are horizontal curves (turning left and right) and vertical curves (driving over a hill or through a valley).

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Engineers typically use a polynomial to describe the shape of vertical curves. The equation of three vertical curves is given. Each represents the shape of the road for a distance of x feet for {x|0 x 400}.

Vertical Curve A: y –0.000 112 5x2 0.04x 100 Vertical Curve B: y 0.000 112 5x2 0.02x 100 Vertical Curve C: y 0.0003x2 0.12x 100

a. What type of polynomial is represented by each equation?

b. By just looking at the equations, which of the curves do you expect to represent similar driving conditions? Explain your choice.

c. Graph each of the functions to check your response to question b.

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Hint: Using a domain of 0 x 400 and a range of 0 y 200 will work well for these functions. Use graph paper and tape in place.

d. Describe the end behaviour you would expect to see for a polynomial used to make a road over a hill and the end behaviour you would expect to see used to make a road through a valley.

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Lesson 2: Modeling Data with Lines of Best Fit1. Determine the independent and dependent variables for each relationship. Justify your

reasoning.

a. The number of hours you practise your favourite sport is related to your skill at that sport.

b. The size of a family’s house is related to the number of people who can comfortably live in the home.

c. The number of hours of darkness is related to the time of year.

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2. A line of best fit has been drawn on the following scatter plot.

a. Describe the characteristics of the line of best fit.

b. Use the line of best fit to estimate the value of y when x 10. Are you using interpolation or extrapolation? Explain.

c. Use the line of best fit to estimate the value of x when y 25. Are you using interpolation or extrapolation? Explain.

d. Use the line of best fit to estimate the value of y when x 30. Are you using interpolation or extrapolation? Explain.

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3. Snow Peak Rentals runs snowmobile tours in Golden, British Columbia. The owner notices that the number of tours run in a season is related to the centimetres of snowfall. The following table shows the historical data for the average snowfall in centimetres as well as the number of tours run each season.

Year 2004 2005 2006 2007 2008 2009 2010 2011

Snowfall (cm) 312 332 285 345 321 365 255 317

Number of Tours 26 28 25 30 28 32 23 27

a. Create a scatter plot to compare the number of snowmobile tours with the amount of snowfall. Use graph paper and tape in place.

b. Describe the trend you see in the data.

c. State an appropriate domain and range for this question.

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d. Determine the equation of the linear regression line using technology. Make sure to round the m- and b-values to the nearest hundredth.

e. What do the slope and y-intercept represent in the context of this problem?

f. Using your equation for the best fit line, predict how many tours Snow Peak Rentals would have if the snowfall was 300 cm.

g. Using your equation for the best fit line, predict how much snowfall there was if Snow Peak Rentals had 33 tours.

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4. Sarah researches the world-record time for the women’s 100 m sprint and creates a scatter plot.

a. Using the two points provided on the line of best fit, find the equation of the line in slope

y-intercept form. Make sure to round to the nearest hundredth for both m and b.

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b. State an appropriate domain and range for this problem.

c. Using your model (i.e., the equation of the line of best fit), estimate what year the world-record time would be 10.75 s.

d. Using your model, what would the world-record time be after 70 years?

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Lesson 3: Modeling Data with a Curve of Best Fit1. Liam was practicing his shot put throws for an upcoming track meet. His coach took

pictures of the ball with a time-lapse camera and measured the height of the ball versus time since the throw.

The coach was hoping this information could help Liam to improve his throwing ability. The coach recorded the information in the following chart:

Time (s) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Height (cm) 152 272 358 402 405 366 297 181

a. What would be the most appropriate regression model to use in this situation, explain your reasoning?

b. State an appropriate domain and range. What does each represent?

c. Find the regression equation.

d. Use your equation to find the height of the ball after 0.5 s.

e. How long would it take for the ball to hit the ground?

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2. Celiac disease is a digestive disease that damages the small intestine and affects the absorption of food. People who have this disease cannot tolerate gluten, which is a protein found in many grain products such as wheat, barley, and rye. In recent years, the incidence of diagnosed cases of this disease has risen dramatically. The following data has been collected on the incidence of celiac disease since 1950.

Years Since 1950 0 8 16 24 32 40 48 52 56 60

Diagnosed Cases of Celiac

100010 11 14 13 11 9 12 18 31 40

a. Create a scatter plot of this data. Use graph paper and tape in place.

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b. What regression model would best fit this model; explain your reasoning?

c. Find an appropriate equation for the curve of best fit.

d. Using your model, how many cases of celiac disease may be diagnosed in 2020?

e. Using your model, in what year(s) were there 10 500 diagnosed cases?

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3. For each of the following graphs, you are provided with the scatter plot and the line or curve of best fit.

i. Devise a scenario that could be modelled using the data. Don’t use a scenario that has already been provided in the textbook or in the lesson. Provide the appropriate domain and range for your scenario.

ii. Make up a question that would involve interpolation and the use of the model (regression equation) to solve. Provide the solution.

iii. Make up a question that would involve extrapolation and the use of the model (regression equation) to solve. Provide the solution.

a. The equation of the line of best fit is y 1.15x 4.42.

b. The equation of the quadratic curve of best fit is y 0.32x2 4.88x 2.79.

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c. The equation of the cubic curve of best fit is y 0.037x3 0.79x2 4.74x 4.27.

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Module Four SUMMARYIn the Module 4 Project you explored and analyzed shapes of polynomials found in our environment.

Following are some of the key ideas you learned in each lesson.

Lesson 1 The general shape of the graph of a polynomial function is determined by the type of function (linear, quadratic, cubic) and the sign of the leading coefficient.

Linear and Cubic, Positive Leading Coefficient

Linear and Cubic, Negative Leading Coefficient

Quadratic, Positive Leading Coefficient

Quadratic, Negative Leading Coefficient

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Lesson 2 Lines of best fit can be created for data that exhibits a linear trend.

Lesson 3 Curves of best fit can be created for data that exhibits quadratic or cubic trends.

Quadratic Trend

Cubic Trend

Mathematics 30-2 Learn EveryWare © 2012 Alberta EducationLast modified: Monday, 14 January 2013, 10

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MODULE 4 – POLYNOMIALS SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach questions and study notes securely to this booklet before you hand everything in.

Text: Principles of Mathematics 12 – Chapter 5 POLYNOMIAL FUNCTIONS

Section 5.1: Page 277, #1, 2, 3abc

Section: 5.2: Pages 387 to 291, #5abde, 7abc, 10

Section: 5.3: Pages 301 to 306, #4, 6, 12

Section: 5.4: Pages 314 to 317, #5, 7

Module 4 is now complete.

Once you have received your corrected work, review your instructor’s comments and prepare for your module four test.