Lesson Increments of a quadratic function form an arithmetic progression

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Increments of a quadratic function form an arithmetic progression


It was shown in the lesson  Free fall and arithmetic progressions  that the distances the  free falling body  falls through in each single second form an  arithmetic progression.
In the lesson  Uniformly accelerated motions and arithmetic progressions  the step forward was made: it was shown that the distances the  uniformly accelerated body  travels
in each single second form an arithmetic progression.
In  THIS  lesson the next step forward is made.

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|    Let  y=f%28x%29  be an arbitrary quadratic function with the quadratic polynomial  f%28x%29+=+ax%5E2+%2Bbx+%2B+c.
|    Let suppose that the uniform sequence of points along the X-axis is given x%5B1%5D, x%5B2%5D, x%5B3%5D, . . . , x%5Bn%5D, . . . with the constant step  dx  so that
|            x%5Bk%5D-x%5Bk-l%5D = dx = const for any two sequential terms  x%5Bk-l%5D,  x%5Bk%5D.
|    Let suppose in addition that the entire sequence x%5B1%5D, x%5B2%5D, x%5B3%5D, . . . , x%5Bn%5D, . . . belong to the monotonicity range of the quadratic function f%28x%29.
|    Consider the increments dy%5Bk%5D = f%28x%5Bk%2B1%5D%29+-+f%28x%5Bk%5D%29, k = 1, 2, . . . , n, . . . .
|    Then the sequence  dy%5B1%5D, dy%5B2%5D, dy%5B3%5D, . . ., dy%5Bn%5D, . . . is the arithmetic progression.
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It is clear that this statement generalizes that of the above mentioned lessons.

Before proving the statement, let us check it for the first terms of the sequence by making direct calculations.

Let us take, for instance, the quadratic polynomial  f%28x%29 = x%5E2+-+2x+-+3.
The plot of this polynomial  y = x%5E2+-+2x+-3 is shown in the Figure  below.

I calculated the values of the polynomial at  x = 1, 2, 3, 4, 5 and 6  and placed the results to the  Table.  You can calculate the values yourself.

The first column in this  Table  contains the values of x.  The second column contains the corresponding values of y = x%5E2+-+2x+-3.  The third column contains
the increments  s%5Bk%5D  that are equal to  y(k+1) - y(k):  s%5Bk%5D = y%5Bk%2B1%5D+-+y%5Bk%5D.  The numbers in the third column are exactly that of the sequence introduced at the beginning
of this lesson.  The statement relates to these numbers and says that they form the arithmetic progression.  At the end, the fourth column contains the differences
between consecutive terms s%5Bn%5D of the third column. As you see, numbers in the fourth column are all the same, which means that the sequence  s%5B1%5D,  s%5B2%5D,  s%5B3%5D,  s%5B4%5D,  s%5B5%5D
is the arithmetic progression. The  Figure  show all the introduced data.


Table.  Quadratic function and its increments

  Values of x
        1
        2
        3
        4
        5
        6



















        Values of y=f(x)
             -4.00
             -3.00
               0.00
               5.00
              12.00
              21.00



















      s%5Bn%5D = y%28n%2B1%29-y%28n%29, increments of y
              1.00
              3.00
              5.00
              7.00
              9.00




















      s%5Bn%2B1%5D+-+s%5Bn%5D
        2.00
        2.00
        2.00
        2.00





















    


    
Figure. Plot of the quadratic function y+=+x%5E2+-2x-3

Now we will prove the statement in general terms.

The value  f%28x%5Bk%5D%29  of the quadratic polynomial  f%28x%29 = a%2Ax%5E2+%2B+b%2Ax+%2B+c  at  x=x%5Bk%5D  is

f%28x%5Bk%5D%29 = a%2Ax%5Bk%5D%5E2+%2B+b%2Ax%5Bk%5D+%2B+c.

The value  f%28x%5Bk%2B1%5D%29  of the quadratic polynomial  f%28x%29 = a%2Ax%5E2+%2B+bx+%2B+c  at  x=x%5Bk%2B1%5D+=+x%5Bk%5D%2Bdx  is

f%28x%5Bk%2B1%5D%29 = a%2Ax%5Bk%2B1%5D%5E2+%2B+b%2Ax%5Bk%2B1%5D+%2B+c = a%2A%28x%5Bk%5D%2Bdx%29%5E2+%2B+b%2A%28x%5Bk%5D%2Bdx%29+%2B+c = a%2Ax%5Bk%5D%5E2+%2B+2a%2Ax%5Bk%5D%2Adx+%2B+a%2Adx%5E2+%2B+b%2Ax%5Bk%5D+%2B+b%2Adx+%2B+c.

The increment  s%5Bk%5D = f%28x%5Bk%2B1%5D%29+-+f%28x%5Bk%5D%29  is equal to
s%5Bk%5D = a%2Ax%5Bk%5D%5E2+%2B+2a%2Ax%5Bk%5D%2Adx+%2B+a%2Adx%5E2+%2B+b%2Ax%5Bk%5D+%2B+b%2Adx+%2B+c - %28a%2Ax%5Bk%5D%5E2+%2B+b%2Ax%5Bk%5D+%2B+c%29

Simplify this expression by opening the brackets and canceling the like terms. You get

s%5Bk%5D = 2a%2Ax%5Bk%5D%2Adx + a%2Adx%5E2 + b%2Adx.

We want to prove that the quantities  s%5Bk%5D = 2a%2Ax%5Bk%5D%2Adx + dx%5E2 + b%2Adx  form the arithmetic progression. To do it take the difference  s%5Bk%5D  and  s%5Bk-l%5D. It is equal to
s%5Bk%5D - s%5Bk-l%5D = 2a%2Ax%5Bk%5D%2Adx + a%2Adx%5E2 + b%2Adx - 2a%2Ax%5Bk-l%5D%2Adx - a%2Adx%5E2 - b%2Adx = 2a%2Ax%5Bk%5D%2Adx - 2a%2Ax%5Bk-1%5D%2Adx = 2a%2A%28x%5Bk%5D-x%5Bk-1%5D%29%2Adx = 2a%2Adx%5E2.

Thus, the difference of each two consecutive terms of the sequence  s%5Bn%5D  is the constant value. This means that the sequence  s%5Bn%5D  is the arithmetic progression.
The statement is proved.


Summary
The increments of a quadratic polynomial function over the sequence of uniformly distributed points form an arithmetic progression.


In the example above the sequence  x%5B1%5D=1,  x%5B1%5D=2,  x%5B1%5D=3, . . .  was the sequence of integer numbers. It does not matter for the statement validity that the numbers
 x%5Bk%5D  are integer. The statement is valid for any sequence of real numbers  x%5Bk%5D providing the uniform step  dx = x%5Bk%5D-x%5Bk-l%5D = const. What does matter is the fact that the
entire sequence of real numbers  x%5Bk%5D  belong to the range of monotonicity of the quadratic function  f%28x%29.


My lessons on arithmetic progressions in this site are
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Chocolate bars and arithmetic progressions
    - Free fall and arithmetic progressions
    - Uniformly accelerated motions and arithmetic progressions
    - Increments of a quadratic function form an arithmetic progression                                           (this lesson)
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions
    - Calculating partial sums of arithmetic progressions
    - Finding number of terms of an arithmetic progression
    - Inserting arithmetic means between given numbers
    - Advanced problems on arithmetic progressions
    - Interior angles of a polygon and Arithmetic progression
    - Math Olympiad level problems on arithmetic progression
    - Problems on arithmetic progressions solved MENTALLY
    - Entertainment problems on arithmetic progressions
    - Mathematical induction and arithmetic progressions
    - Mathematical induction for sequences other than arithmetic or geometric

OVERVIEW of my lessons on arithmetic progressions with short annotations is in the lesson  OVERVIEW of lessons on arithmetic progressions.

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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