Inscribed parabola

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Unparable

In school mathematics, the term inscribed parabola describes a construction from the field of geometry in connection with quadratic functions .

definition

An arc of a quadratic function is inscribed in an isosceles triangle ABC with height h and base side length c in such a way that the two sides of the triangle touch the parabolic arc in A and B. Then this parabola is called an inscribed parabola or - in analogy to the inscribed circle - an unparabola .

Note: The parabola in its property as a graph of a quadratic function is a special case of a quadratic Bézier curve .

property

The vertex of an inparabola always halves the height of the associated isosceles triangle ABC.

proof

The parabola has the function equation with the vertex and the derivative . ${\ displaystyle f (x) = ax ^ {2} + d}$ ${\ displaystyle S (0 | d)}$ ${\ displaystyle f '(x) = 2ax}$

From initially follows: ${\ displaystyle f \ left ({\ frac {c} {2}} \ right) = 0 \ Leftrightarrow {\ frac {ac ^ {2}} {4}} + d = 0 \ Leftrightarrow d = - {\ frac {ac ^ {2}} {4}}}$ ${\ displaystyle f (x) = ax ^ {2} - {\ frac {ac ^ {2}} {4}} = a \ left (x ^ {2} - {\ frac {c ^ {2}} { 4}} \ right)}$

The slope of the straight line through and is and is identical to . ${\ displaystyle A \ left (- {\ frac {c} {2}} \ right | 0)}$ ${\ displaystyle C (0 | h)}$ ${\ displaystyle m = {\ frac {h-0} {0- \ left (- {\ frac {c} {2}} \ right)}} = {\ frac {2h} {c}}}$ ${\ displaystyle f '\ left (- {\ frac {c} {2}} \ right) = 2a \ cdot \ left (- {\ frac {c} {2}} \ right) = - ac}$

Therefore , so . ${\ displaystyle {\ frac {2h} {c}} = - ac}$ ${\ displaystyle a = - {\ frac {2h} {c ^ {2}}}}$

Consequently, the c and h-dependent parabolic equation is: . ${\ displaystyle f (x) = \ left (- {\ frac {2h} {c ^ {2}}} \ right) \ left (x ^ {2} - {\ frac {c ^ {2}} {4 }} \ right) = - {\ frac {2h} {c ^ {2}}} x ^ {2} + {\ frac {h} {2}}}$

The vertex of the parabola is therefore and thus halves the height of the isosceles triangle ABC. ${\ displaystyle S \ left (0 | {\ frac {h} {2}} \ right)}$

Individual evidence

↑ Wolfgang Göbels: Inscribed and enveloping parables. German Association for the Promotion of Mathematics and Science Education 63/3 (April 15, 2010) pp. 152–154, ISSN 0025-5866, © Verlag Klaus Seeberger, Neuss.
↑ Inscribed and enveloping parables with Interactive Notes (from Texas Instruments Education Technology - Mathematics Working Group)

Web links

Wolfgang Göbels: Modeling curves with GeoGebra , page 15 , from Raabits

[1] Wolfgang Göbels: Inscribed and enveloping parables. German Association for the Promotion of Mathematics and Science Education 63/3 (April 15, 2010) pp. 152–154, ISSN 0025-5866, © Verlag Klaus Seeberger, Neuss.

[2] Inscribed and enveloping parables with Interactive Notes (from Texas Instruments Education Technology - Mathematics Working Group)