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Posers and Puzzles

Posers and Puzzles

  1. Standard member talzamir
    Art, not a Toil
    31 Oct '11 08:22
    The graph of f(x) = ax^2 + bx + c is a parabola if a does not equal zero. The effect of a and c on the graph are fairly simple, a can flip it upside down and make it tighter or wider, c moves the apex in the direction of the y-axis.

    b is less obvious. It would seem to move the apex of the parabola along a curve, but what curve?
  2. 31 Oct '11 12:17
    Originally posted by talzamir
    The graph of f(x) = ax^2 + bx + c is a parabola if a does not equal zero. The effect of a and c on the graph are fairly simple, a can flip it upside down and make it tighter or wider, c moves the apex in the direction of the y-axis.

    b is less obvious. It would seem to move the apex of the parabola along a curve, but what curve?
    Hint:
    It\'s not, strictly speaking, a curve.


    Richard
  3. Standard member talzamir
    Art, not a Toil
    31 Oct '11 13:33
    Interesting..
    Far as I can tell, all the possible points where the apex can be, if a and c are, are indeed on the same curve.
  4. 31 Oct '11 22:23 / 2 edits
    y = ax^2 + bx + c

    dy/dx = 2ax + b

    At the apex this is zero

    2ax + b = 0

    x = -b/(2a) at the apex

    y = ax^2 + bx + c

    at the apex:

    y = b^2/(4a) - b^2/(2a) + c

    y = -b^2/(4a) + c

    Since the intercept y scales with b^2 and the intercept x scales with b, b moves the intercept along a parabola
  5. Standard member talzamir
    Art, not a Toil
    01 Nov '11 06:38 / 1 edit
    Indeed it does - nicely done.

    Putting it into a more familiar format of y(x) instead of the paired equations of y(b) and x(b).

    Coordinates of the apex being, as you solved it,

    x = -b/(2a)
    y = -b^2/(4a) + c

    Squaring the first equation gives x^2 = b^2/(4a^2).

    Tweaking the second equation and then substituting x into it gives

    y = -b^2/(4a) + c = -a * b^2 / (4a^2) + c = -ax^2 + c

    So it seems not just any parabola, but identical in shape with the original, upside down, and has the apex at (0,c) which is where all the parabolas y = ax^2 + bx + c pass through the y-axis.
  6. 03 Nov '11 13:08
    Originally posted by iamatiger
    Since the intercept y scales with b^2 and the intercept x scales with b, b moves the intercept along a parabola
    It seems I did not take enough samples... I thought it was a line! Wrongly, apparently.

    Richard
  7. Standard member forkedknight
    Defend the Universe
    03 Nov '11 14:21
    Originally posted by Shallow Blue
    It seems I did not take enough samples... I thought it was a line! Wrongly, apparently.

    Richard
    A line is a curve, anyway.