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Консультация № 110681
23.11.2007, 16:40
0.00 руб.
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нужно составить уравнение верхней ветви параболы по рисунку. http://up.li.ru/?id=322341;%EF%E0%F0%E0%E1%EE%EB%E0.bmp
вершина в точке (8,3) пересекает ось оу в точке (0,5)
у меня получилось (y-5)^2=-2p(x-8)
вопрос не могу найти p, и как показать что именно верхняя ветвь параболы
вершина в точке (8,3) пересекает ось оу в точке (0,5)
у меня получилось (y-5)^2=-2p(x-8)
вопрос не могу найти p, и как показать что именно верхняя ветвь параболы
Обсуждение
Неизвестный
23.11.2007, 19:09
общий
это ответ
Здравствуйте, lyalya!
Уравнение в общем виде параболы, "лежащей на боку" x = ay² + by + c
Вершина параболы (определяется аналогично тому, как это делается для обычной параболы - нахождением экстремума x‘=0) - это точка с координатами (x,y), где y=-b/(2a)
Итак, парабола проходит через точку (8,3) => 8 = a*3² + b*3 + c = 9a + 3b + c
Это вершина => 3 = -b/(2a)
Парабола проходит через точку (0,5) => 0 = a*5² + b*5 + c = 25a + 5b + c
Система из трех уравнений с тремя неизвестными => находим a = -2, b = 12, c = -10
x = -2y² + 12y + -10
Выражаем отсюда y (решаем квадратное уравнение 2y² - 12y + 10 + x = 0).
y = 3 ± √(16-2x)/2
Уравнение y = 3 + √(16-2x)/2 - это верхняя ветвь параболы
Уравнение y = 3 - √(16-2x)/2 - это нижняя ветвь параболы
Уравнение в общем виде параболы, "лежащей на боку" x = ay² + by + c
Вершина параболы (определяется аналогично тому, как это делается для обычной параболы - нахождением экстремума x‘=0) - это точка с координатами (x,y), где y=-b/(2a)
Итак, парабола проходит через точку (8,3) => 8 = a*3² + b*3 + c = 9a + 3b + c
Это вершина => 3 = -b/(2a)
Парабола проходит через точку (0,5) => 0 = a*5² + b*5 + c = 25a + 5b + c
Система из трех уравнений с тремя неизвестными => находим a = -2, b = 12, c = -10
x = -2y² + 12y + -10
Выражаем отсюда y (решаем квадратное уравнение 2y² - 12y + 10 + x = 0).
y = 3 ± √(16-2x)/2
Уравнение y = 3 + √(16-2x)/2 - это верхняя ветвь параболы
Уравнение y = 3 - √(16-2x)/2 - это нижняя ветвь параболы
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