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Lesson 7.1

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In Chapter 1, you studied arithmeticsequences, which have a common difference

between consecutive terms. This commondifference is the slope of a line through thegraph of the points. So, if you choose x-values along the line that form an arithmetic

sequence, the corresponding y-values willalso form an arithmetic sequence.

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You have also studied several kinds ofnonlinear sequences and functions, which donot have a common difference or a constantslope.

In this lesson you will discover that evennonlinear sequences sometimes have aspecial pattern in their differences.

These patterns are often described bypolynomials.

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When a polynomial is set equal to a secondvariable, such as y, it defines a polynomialfunction.

The degree of a polynomial or polynomialfunction is the power of the term that has thegreatest exponent. Linear Functions: y=3-2x

largest power of x is 1 Cubic Function: y=2x3+1x2-2x+3

largest power of x is 3 If the degrees of the terms of a polynomial

decrease from left to right, the polynomial is ingeneral form.

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A polynomial that has only one term is calleda monomial.

A polynomial with two terms is a binomial

A polynomial with three terms is atrinomial

. Polynomials with more than three terms are

usually just called polynomials.

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For the 2nd-degree polynomial function, the D2values are

constant, and for the 3rd-degree polynomial function, the D3

values are constant. What do you think will happen with a 4th-

or 5th-degree polynomial function?

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Find a polynomial function that models therelationship between the number of sides andthe number of diagonals of a convex polygon.Use the function to find the number of

diagonals of a dodecagon (a 12-sidedpolygon).

You need to create a table of values withevenly spaced x-values. Sketch polygons withincreasing numbers of sides. Then draw all oftheir diagonals.

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Let x be the number of sides and y be the number of

diagonals.

You may notice a pattern in the number of diagonals

that will help you extend your table beyond the

sketches you make.

Calculate the finite differences to determine thedegree of the polynomial function.

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You can stop finding differences when the values of a set of

differences are constant. Because the values of D2are

constant, you can model the data with a 2nd-degree

polynomial function like y =ax2+ bx + c.

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16 4 2

36 6 9

64 8 20

a b c

a b c

a b c

16 4 1

36 6 1

64 8 1

a

b

c

2

9

20

=

16 4 1

36 6 1

64 8 1

a

b

c

2

9

20

=

116 4 1

36 6 1

64 8 1

116 4 1

36 6 1

64 8 1

a

b

c

0.5

1.5

0

=

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What function modelsthe height of anobject falling due tothe force of gravity?

Use a motion sensorto collect data, andanalyze the data tofind a function.

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1. Set the sensor to collect distance dataapproximately every 0.05 s for 2 to 5 s.

2. Place the sensor on the floor. Hold a smallpillow at a height of about 2 m, directlyabove the sensor.

3. Start the sensor and drop the pillow.

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Create scatter plots of the original data (time,height), then a scatter plot of (time, firstdifference), and finally a scatter plot of (time,second difference).

Write a description of each graph from theprevious step and what these graphs tell youabout the data.

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The graph of (time, height)

appears parabolic and suggests

that the correct model may be a

quadratic (2nd-degree) polynomialfunction.

The graph of (time, first difference)

shows that the first differences are

not constant; because they

decrease in a linear fashion, the

second differences are likely to be

constant.

The graph of (time, second

difference) shows that the seconddifferences are nearly constant, so

the correct model should be a 2nd-

degree polynomial function.

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Follow the example on page 380 to write asystem of three equations in three variablesfor your data. Solve your system to find anequation to model the position of a free-

falling object dropped from a height of 2 m.

Systems will vary depending on the points

chosen; the function should be approximately

equivalent to y=-4.9x2+2.

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Use the finite differences method to find thedegree of the polynomial function thatmodels your data. Stop when the differencesare nearly constant.

There are two ways you can find the finitedifferences: From the table of values

From the homescreen

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First

First name Column C:

diff1 and Column D:

diff2

Then in the gray row

enter the

command list(height)

in column C and

list(diff1) in column D

What do you notice

about the seconddifferences?

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Create scatter plots of the original data (time,height), then a scatter plot of (time, firstdifference), and finally a scatter plot of (time,second difference).

Write a description of each graph from theprevious step and what these graphs tell youabout the data.

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Based on your results from using finitedifferences, what is the degree of thepolynomial function that models free fall?Write the general form of this polynomial

function.

2nd degree: y = ax2+ bx + c

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Follow the example on page 380 to write asystem of three equations in three variablesfor your data. Solve your system to find anequation to model the position of a free-

falling object dropped from a height of 2 m.

Systems will vary depending on the points

chosen; the function should be approximately

equivalent to y=-4.9x2+2.

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Lets select (0,2), (0.2, 1.804) and(0.4, 1.216) as the three points.

Replacing these in the equation y = ax2

+ bx +cwe get 2=0a+0b+c

1.804=0.04a+0.2b+c

1.216=0.16a+0.4b+c

Y=-4.9x2+2