Puzzles
23 December
I draw the parabola and mark points on the parabola at and .
I then draw a straight line connecting these two points.
At which value of does this line intercept the -axis?
6 December
is a quadratic with real coefficients. For all real numbers ,
. What is ?
Between quadratics
is a quadratic polynomial with real coefficients. For all real numbers ,
. Find .
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Therefore the minimum point of both of these quadratics is . will only be between these if:
We know that , so:
Therefore . This means that:
Parabola
On a graph of , two lines are drawn at and (for . The points where these lines intersect the parabola are connected.
What is the y-coordinate of the point where this line intersects the y-axis?
Show answer & extension
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The co-ordinates of the points where the lines intersect the parabola are and . Hence the gradient of the line between them is:
Therefore the y-coordinate is:
Ferdinand Möbius, who discovered this property called the curve a Multiplicationsmaschine or 'multiplication machine' as it could be used to perform multiplication.
Extension
How could you use the graph of to divide 100 by 7?