Lesson Increments of a quadratic function form an arithmetic progression
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<H2>Increments of a quadratic function form an arithmetic progression</H2> It was shown in the lesson <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Free-fall-and-arithmetic-progressions.lesson>Free fall and arithmetic progressions</A> that the distances the <B>free falling body</B> falls through in each single second form an <B>arithmetic progression</B>. In the lesson <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Uniformly-accelerated-motions-and-arithmetic-progressions.lesson>Uniformly accelerated motions and arithmetic progressions</A> the step forward was made: it was shown that the distances the <B>uniformly accelerated body</B> travels in each single second form an <B>arithmetic progression</B>. In <B>THIS</B> lesson the next step forward is made.<BLOCKQUOTE><TABLE> <TD> +------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Let {{{y=f(x)}}} be an arbitrary quadratic function with the quadratic polynomial {{{f(x) = ax^2 +bx + c}}}. | Let suppose that the uniform sequence of points along the X-axis is given {{{x[1]}}}, {{{x[2]}}}, {{{x[3]}}}, . . . , {{{x[n]}}}, . . . with the constant step {{{dx}}} so that | {{{x[k]-x[k-l]}}} = {{{dx}}} = {{{const}}} for any two sequential terms {{{x[k-l]}}}, {{{x[k]}}}. | Let suppose in addition that the entire sequence {{{x[1]}}}, {{{x[2]}}}, {{{x[3]}}}, . . . , {{{x[n]}}}, . . . belong to the monotonicity range of the quadratic function {{{f(x)}}}. | Consider the increments {{{dy[k]}}} = {{{f(x[k+1]) - f(x[k])}}}, k = 1, 2, . . . , n, . . . . | Then <B>the sequence {{{dy[1]}}}, {{{dy[2]}}}, {{{dy[3]}}}, . . ., {{{dy[n]}}}, . . . is the arithmetic progression</B>. +------------------------------------------------------------------------------------------------------------------------------------------------------------------------- </TD> <TD> + | | | | | | | | | | + </TD> </Table> </BLOCKQUOTE>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(x)}}} = {{{x^2 - 2x - 3}}}. The plot of this polynomial {{{y}}} = {{{x^2 - 2x -3}}} is shown in the <B>Figure</B> below. I calculated the values of the polynomial at x = 1, 2, 3, 4, 5 and 6 and placed the results to the <B>Table</B>. You can calculate the values yourself. The first column in this <B>Table</B> contains the values of x. The second column contains the corresponding values of {{{y}}} = {{{x^2 - 2x -3}}}. The third column contains the increments {{{s[k]}}} that are equal to y(k+1) - y(k): {{{s[k]}}} = {{{y[k+1] - y[k]}}}. 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[n]}}} of the third column. As you see, numbers in the fourth column are all the same, which means that the sequence {{{s[1]}}}, {{{s[2]}}}, {{{s[3]}}}, {{{s[4]}}}, {{{s[5]}}} is the arithmetic progression. The <B>Figure</B> show all the introduced data. <B>Table</B>. Quadratic function and its increments <TABLE> <TD> Values of x 1 2 3 4 5 6 </TD> <TD> Values of y=f(x) -4.00 -3.00 0.00 5.00 12.00 21.00 </TD> <TD> {{{s[n]}}} = {{{y(n+1)-y(n)}}}, increments of y 1.00 3.00 5.00 7.00 9.00 </TD> <TD> {{{s[n+1] - s[n]}}} 2.00 2.00 2.00 2.00 </TD> <TD> {{{drawing( 210, 540, -7.0, 2.0, -5.2, 22.8, line ( 0.0, -5.0, 0.0, 23), line (-0.2, -5.0, 0.2, -5.0), line (-0.2, -4.0, 0.2, -4.0), line (-0.2, -3.0, 0.2, -3.0), line (-0.2, -2.0, 0.2, -2.0), line (-0.2, -1.0, 0.2, -1.0), line (-0.2, 0.0, 0.2, 0.0), line (-0.2, 1.0, 0.2, 1.0), line (-0.2, 2.0, 0.2, 2.0), line (-0.2, 3.0, 0.2, 3.0), line (-0.2, 4.0, 0.2, 4.0), line (-0.2, 5.0, 0.2, 5.0), line (-0.2, 6.0, 0.2, 6.0), line (-0.2, 7.0, 0.2, 7.0), line (-0.2, 8.0, 0.2, 8.0), line (-0.2, 9.0, 0.2, 9.0), line (-0.2, 10.0, 0.2, 10.0), line (-0.2, 11.0, 0.2, 11.0), line (-0.2, 12.0, 0.2, 12.0), line (-0.2, 13.0, 0.2, 13.0), line (-0.2, 14.0, 0.2, 14.0), line (-0.2, 15.0, 0.2, 15.0), line (-0.2, 16.0, 0.2, 16.0), line (-0.2, 17.0, 0.2, 17.0), line (-0.2, 18.0, 0.2, 18.0), line (-0.2, 19.0, 0.2, 19.0), line (-0.2, 20.0, 0.2, 20.0), line (-0.2, 21.0, 0.2, 21.0), line (-0.2, 22.0, 0.2, 22.0), locate (-1.1, -3.6, -4), locate (-1.1, -2.6, -3), locate (-1.1, -1.6, -2), locate (-1.1, -0.6, -1), locate (-0.6, 0.4, 0), locate (-0.6, 1.4, 1), locate (-0.6, 2.4, 2), locate (-0.6, 3.4, 3), locate (-0.6, 4.4, 4), locate (-0.6, 5.4, 5), locate (-0.6, 6.4, 6), locate (-0.6, 7.4, 7), locate (-0.6, 8.4, 8), locate (-0.6, 9.4, 9), locate (-1.0, 10.4, 10), locate (-1.0, 11.4, 11), locate (-1.0, 12.4, 12), locate (-1.0, 13.4, 13), locate (-1.0, 14.4, 14), locate (-1.0, 15.4, 15), locate (-1.0, 16.4, 16), locate (-1.0, 17.4, 17), locate (-1.0, 18.4, 18), locate (-1.0, 19.4, 19), locate (-1.0, 20.4, 20), locate (-1.0, 21.4, 21), locate (-1.0, 22.4, 22), line ( 0.4, -4.0, 1.2, -4.0), line ( 0.4, -3.0, 1.2, -3.0), locate ( 0.5, -3.1, s(1)), line ( 0.4, 0.0, 1.2, 0.0), locate ( 0.5, -1.1, s(2)), line ( 0.9, 0.0, 0.9, -0.9), line ( 0.9, -2.1, 0.9, -3.0), line ( 0.4, 5.0, 1.2, 5.0), locate ( 0.5, 2.9, s(3)), line ( 0.9, 5.0, 0.9, 3.1), line ( 0.9, 1.9, 0.9, 0.0), line ( 0.4, 12.0, 1.2, 12.0), locate ( 0.5, 8.9, s(4)), line ( 0.9, 12.0, 0.9, 9.1), line ( 0.9, 7.9, 0.9, 5.0), line ( 0.4, 21.0, 1.2, 21.0), locate ( 0.5, 16.9, s(5)), line ( 0.9, 21.0, 0.9, 17.1), line ( 0.9, 15.9, 0.9, 12.0), line (-5.9, -4.0, -1.4, -4.0), line (-2.3, -3.0, -1.4, -3.0), locate (-2.2, -3.1, d(1)), line (-3.3, 0.0, -1.4, 0.0), locate (-3.2, -1.5, d(2)), line (-2.7, 0.0, -2.7, -1.3), line (-2.7, -2.4, -2.7, -4.0), line (-4.3, 5.0, -1.4, 5.0), locate (-4.2, 1.1, d(3)), line (-3.7, 5.0, -3.7, 1.3), line (-3.7, 0.1, -3.7, -4.0), line (-5.3, 12.0, -1.4, 12.0), locate (-5.2, 4.5, d(4)), line (-4.7, 12.0, -4.7, 4.7), line (-4.7, 3.6, -4.7, -4.0), line (-6.3, 21.0, -1.4, 21.0), locate (-6.2, 8.9, d(5)), line (-5.7, 21.0, -5.7, 9.1), line (-5.7, 7.9, -5.7, -4.0) )}}} </TD> <TD> {{{graph( 300, 540, -3.0, 7.0, -5.0, 23, x^2 -2x -3 )}}} <B>Figure</B>. Plot of the quadratic function {{{y = x^2 -2x-3}}} </TD> </Table> Now we will prove the statement in general terms. The value {{{f(x[k])}}} of the quadratic polynomial {{{f(x)}}} = {{{a*x^2 + b*x + c}}} at {{{x=x[k]}}} is {{{f(x[k])}}} = {{{a*x[k]^2 + b*x[k] + c}}}. The value {{{f(x[k+1])}}} of the quadratic polynomial {{{f(x)}}} = {{{a*x^2 + bx + c}}} at {{{x=x[k+1] = x[k]+dx}}} is {{{f(x[k+1])}}} = {{{a*x[k+1]^2 + b*x[k+1] + c}}} = {{{a*(x[k]+dx)^2 + b*(x[k]+dx) + c}}} = {{{a*x[k]^2 + 2a*x[k]*dx + a*dx^2 + b*x[k] + b*dx + c}}}. The increment {{{s[k]}}} = {{{f(x[k+1]) - f(x[k])}}} is equal to {{{s[k]}}} = {{{a*x[k]^2 + 2a*x[k]*dx + a*dx^2 + b*x[k] + b*dx + c}}} - {{{(a*x[k]^2 + b*x[k] + c)}}} Simplify this expression by opening the brackets and canceling the like terms. You get {{{s[k]}}} = {{{2a*x[k]*dx}}} + {{{a*dx^2}}} + {{{b*dx}}}. We want to prove that the quantities {{{s[k]}}} = {{{2a*x[k]*dx}}} + {{{dx^2}}} + {{{b*dx}}} form the arithmetic progression. To do it take the difference {{{s[k]}}} and {{{s[k-l]}}}. It is equal to {{{s[k]}}} - {{{s[k-l]}}} = {{{2a*x[k]*dx}}} + {{{a*dx^2}}} + {{{b*dx}}} - {{{2a*x[k-l]*dx}}} - {{{a*dx^2}}} - {{{b*dx}}} = {{{2a*x[k]*dx}}} - {{{2a*x[k-1]*dx}}} = {{{2a*(x[k]-x[k-1])*dx}}} = {{{2a*dx^2}}}. Thus, the difference of each two consecutive terms of the sequence {{{s[n]}}} is the constant value. This means that the sequence {{{s[n]}}} is the arithmetic progression. The statement is proved. <B>Summary</B> <B>The increments of a quadratic polynomial function over the sequence of uniformly distributed points form an arithmetic progression</B>. In the example above the sequence {{{x[1]=1}}}, {{{x[1]=2}}}, {{{x[1]=3}}}, . . . was the sequence of integer numbers. It does not matter for the statement validity that the numbers {{{x[k]}}} are integer. The statement is valid for any sequence of real numbers {{{x[k]}}} providing the uniform step {{{dx}}} = {{{x[k]-x[k-l]}}} = const. What does matter is the fact that the entire sequence of real numbers {{{x[k]}}} belong to the range of monotonicity of the quadratic function {{{f(x)}}}. My lessons on arithmetic progressions in this site are - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Chocolate-bars-and-arithmetic-progressions.lesson>Chocolate bars and arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Free-fall-and-arithmetic-progressions.lesson>Free fall and arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Uniformly-accelerated-motions-and-arithmetic-progressions.lesson>Uniformly accelerated motions and arithmetic progressions</A> - Increments of a quadratic function form an arithmetic progression (this lesson) - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson>Solved problems on arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Calculating-partial-sums-of-arithmetic-progressions.lesson>Calculating partial sums of arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Finding-number-of-terms-of-an-arithmeti--progression.lesson>Finding number of terms of an arithmetic progression</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Inserting-arithmetic-means-between-given-numbers.lesson>Inserting arithmetic means between given numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Advanced-problems-on-arithmetic-progressions.lesson>Advanced problems on arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Interior-angles-of-a-polygon-and-Arithmetic-progression.lesson>Interior angles of a polygon and Arithmetic progression</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Marh-Olimpiad-level-problem-on-arithmetic-progression.lesson>Math Olympiad level problems on arithmetic progression</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions-solved-MENTALLY.lesson>Problems on arithmetic progressions solved MENTALLY</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Entertainment-problems-on-arithmetic-progressions.lesson>Entertainment problems on arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A> OVERVIEW of my lessons on arithmetic progressions with short annotations is in the lesson <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Review-of-lessons-on-arithmetic-progressions.lesson>OVERVIEW of lessons on arithmetic progressions</A>. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
