Fun\303\247\303\243o do Segundo Grau:V\303\251rtice de uma Par\303\241bola<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Motiva\303\247\303\243o</Font></Text-field>O que um farol de carro e uma antena parab\303\263lica t\303\252m em comum?Ambos s\303\243o dispositivos que s\303\243o constru\303\255dos sobre as propriedades \303\272nicas de uma par\303\241bola. A par\303\241bola \303\251 uma curva sim\303\251trica, e sobre o seu eixo de simetria h\303\241 um ponto especial chamado o 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"raio de luz " que emana do foco vai acertar a par\303\241bola e ser refletido paralelamente ao eixo de simetria, como nos far\303\263is de um carro. Por outro lado, em uma antena parab\303\263lica, r\303\241dio ou ondas de som parecem um parabol\303\263ide (a forma 3-D que voc\303\252 obt\303\251m girando uma par\303\241bola em torno do seu eixo) s\303\243o refletidas para o receptor localizado no foco.
Voc\303\252 pode pensar em outros dispositivos que s\303\243o constru\303\255dos sobre as propriedades \303\272nicas de uma par\303\241bola?<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">O Que \303\251 uma Par\303\241bola?</Font></Text-field>Esta se\303\247\303\243o d\303\241 a explica\303\247\303\243o geom\303\251trica de uma par\303\241bola.
Cada ponto na curva chamada par\303\241bola \303\251 equidistante de um ponto fixo denominado o foco, e de uma reta fixa chamada de diretriz. A ponto de uma par\303\241bola que est\303\241 mais pr\303\263ximo da reta (e, portanto, eq\303\274idistante da reta e do foco) \303\251 o v\303\251rtice. (Ver Figura 1).Se a diretriz \303\251 uma reta horizontal, a par\303\241bola pode ser descrito por uma equa\303\247\303\243o quadr\303\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Onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlElJm5lO0YnRjkvRj1RJ25vcm1hbEYnLyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVS1JI21uR0YkNiVRIjBGJ0Y5RkNGOUZDRitGOUZDRitGOUZD. Para tal par\303\241bola, o v\303\251rtice \303\251 o ponto mais baixo (a > 0) ou o ponto mais alto (a < 0) na par\303\241bola. No primeiro caso, a par\303\241bola abre para cima, no segundo, para baixo.
A Figura 1 mostra uma par\303\241bola com a abertura para cima com o v\303\251rtice na origem.
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QVJFTkNZRzYjJCIiISEiIi0lK0FYRVNMQUJFTFNHNiUtSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJ4NiIvJSdmYW1pbHlHUShERUZBVUxUNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiUSE2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNS0lKFNDQUxJTkdHNiMlLENPTlNUUkFJTkVERy0lKV9WSVNJQkxFRzYjIiIiLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkUyIhIiItJSlCT1VORFNfWUc2IyQiJHEqISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlIXkkISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJWc/ISIiLSUpQ0hJTERSRU5HNiItJStBTk5PVEFUSU9ORzYnLSUpQk9VTkRTX1hHNiMkIiRJIiEiIi0lKUJPVU5EU19ZRzYjJCIkKykhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVnSSEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiVxOyEiIi0lKUNISUxEUkVORzYiRzYiLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Figura 1 A par\303\241bola 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 \303\251 a curva vermelha. A diretriz \303\251 a linha verde, o foco \303\251 o ponto vermelho, eo v\303\251rtice \303\251 o ponto azul.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Forma V\303\251rtice de uma Par\303\241bola </Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=A forma geral da par\303\241bola descrita \303\251 uma equa\303\247\303\243o quadr\303\241tica 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. A forma padr\303\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\303\251 tamb\303\251m chamada de forma v\303\251rtice, estando o v\303\251rtice est\303\241 no ponto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJrRidGNEY3RjpGN0ZBRjdGQUY3RkE=.Aten\303\247\303\243o: A equa\303\247\303\243o para a forma v\303\251rtice de uma par\303\241bola tem um sinal de menos (-) na frente de h. Assim, para escrever o v\303\251rtice, voc\303\252 escreve o n\303\272mero subtra\303\255do de x, mais o n\303\272mero adicionado ao final da equa\303\247\303\243o. Veja o Exemplo 1.<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Demonstra\303\247\303\243o</Font></Text-field>Explore uma par\303\241bola e seu v\303\251rtice, ajustando as barras deslizantes para o \303\242ngulo e velocidade.
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
end use;
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