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Characterizatio parabolarvm earvmqve delineationes

Svmmarivm

Parabolam characterizare est ex eivs aeqvatione ea data colligere qvae ad eam delineandam svfficivnt: plagam in qvam aperitvr, verticem, axem symmetriae, focvm, directricem et intersectiones cvm axibvs coordinatarvm. Haec lectio ostendit qvomodo haec data ex forma verticis et ex forma polynomiali obtineantvr, formvlas generales verticis, foci et directricis ex coefficientibvs expressas dedvcit, intersectiones cvm axe xx cvm discriminante conivngit et symmetria cvrvae vtitvr ad calcvlos breviandos. Conclvditvr certa ratione ad delineationem advmbrandam.

Proposita discendi

  1. Verticem, focvm, directricem et axem parabolae in forma verticis vel in forma polynomiali datae obtinere.
  2. Qvadratvm complendo formvlas generales elementorvm parabolae y=ax2+bx+cy = ax^2 + bx + c dedvcere.
  3. Intersectiones cvm axibvs determinare et per discriminans diivdicare vtrvm parabola axem xx secet.
  4. Symmetria respectv axis vti ad verticem inveniendvm et delineationem ex notitiis obtentis advmbrare.

Dvae formae aeqvationis

In lectione de aeqvatione parabolarvm visvm est parabolam axe verticali, vertice in (h,k)(h, k), foco (h,k+f)(h, k + f) et directrice y=k−fy = k - f, cvm f≠0f \neq 0, aeqvationi satisfacere qvam hic formam verticis appellabimvs; valor absolvtvs ipsivs ff est distantia focalis, et signvm indicat in qvam plagam cvrva aperiatvr:

(x−h)2=4f(y−k)(x - h)^2 = 4f(y - k)

Demonstratvm est etiam, in theoremate de delineationibvs polynomiorvm qvadraticorvm, hanc aeqvationem, explicata yy, in polynomivm qvadraticvm converti et, vice versa, delineationem cvivsqve polynomii qvadratici esse parabolam axe verticali. Formam polynomialem appellabimvs aeqvationem

y=ax2+bx+c,a≠0y = ax^2 + bx + c, \quad a \neq 0

Ambae eandem cvrvam describvnt, sed non eadem notitia ostendvnt: forma verticis geometriam exhibet (verticem, focvm, directricem), forma avtem polynomialis intersectionem cvm axe yy exhibet, qvae est (0,c)(0, c), et intersectiones cvm axe xx aeqvatione qvadratica solvenda compvtare sinit. Parabolam characterizare est haec data colligere, qvibvs eivs delineatio advmbratvr:

  1. Plaga apertvrae: svrsvm vel deorsvm.
  2. Vertex et axis symmetriae.
  3. Focvs et directrix.
  4. Intersectiones cvm axe xx, si exstant, et intersectio cvm axe yy.