Enveloping parabola

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Umparabola (red) and unparabola (green)

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

definition

Let an arc of a quadratic function be described for an isosceles triangle ABC with height h and base side length c in such a way that it runs through the three corner points A, B and C. Then this parabola is called the enveloping parabola or - in analogy to the circumference - umparabola .

properties

The area dimension of the umparabolic arch is twice as large as that of the inparable arch.
The area dimensions of the inconsistent arch , the triangle ABC and the reversible arch are related to each other in this order as 2: 3: 4.

proof

The Archimedes' formula for parabolic segments is used, which Archimedes proves in one of his main writings with the title Quadrature of the Parabola , Latin De quadratura parabolae . This formula says that the area of a parabolic segment with the base g and the height h has the dimension number . ${\ displaystyle \ textstyle {\ frac {2} {3}}}$ ${\ displaystyle \ mathrm {gh}}$

If, according to the Archimedes' formula, one calculates the dimensions of the patches, which include the inparabola and the umparabola with the x-axis, one obtains for the

Inparable (see property there ) and for the umparable . Consequently, the following applies: . ${\ displaystyle A_ {I} = {\ frac {2} {3}} c \ cdot {\ frac {h} {2}} = {\ frac {1} {3}} c \ cdot h}$ ${\ displaystyle A_ {U} = {\ frac {2} {3}} c \ cdot h}$ ${\ displaystyle A_ {U} = 2 \ cdot A_ {I}}$

So the area dimension of the umparabolic arch is twice as large as that of the unparable arch.

Since the area dimension of the isosceles triangle is known to be , the ratio equation applies to the area dimensions, or to put it more simply: ${\ displaystyle A_ {D} = {\ frac {1} {2}} \ cdot c \ cdot h}$ ${\ displaystyle A_ {I}: A_ {D}: A_ {U} = {\ frac {1} {3}}: {\ frac {1} {2}}: {\ frac {2} {3}} }$

${\ displaystyle A_ {I}: A_ {D}: A_ {U} = 2: 3: 4}$ .

Individual evidence

↑ Wolfgang Göbels: Inscribed and enveloping parables. (PDF; 71 kB) In: Mathematics and science lessons. Volume 63, No. 3 (April 15, 2010) German Association for the Promotion of Mathematics and Science Education , ISSN 0025-5866 , © Verlag Klaus Seeberger, Neuss. Pp. 152-154
↑ Inscribed and enveloping parables with Interactive Notes (from Texas Instruments Education Technology - Mathematics Working Group)
↑ Outlook on the quadrature of the parabola in Archimedes' (PDF; 12 MB), publication by Oliver Deiser, professor at the Technical University of Munich

Web links

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

[1] Wolfgang Göbels: Inscribed and enveloping parables. (PDF; 71 kB) In: Mathematics and science lessons. Volume 63, No. 3 (April 15, 2010) German Association for the Promotion of Mathematics and Science Education , ISSN 0025-5866 , © Verlag Klaus Seeberger, Neuss. Pp. 152-154

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

[3] Outlook on the quadrature of the parabola in Archimedes' (PDF; 12 MB), publication by Oliver Deiser, professor at the Technical University of Munich