terça-feira, outubro 18, 2011

08 - FUNÇÃO DO 2º GRAU

FUNÇÃO DO 2º GRAU


DEFINIÇÃO

Chama-se função quarática a função definida por:


Y = ax² + bx + c

onde a, b e c são números reais e a ≠ 0.

Exemplos:


1)     Y = x² -  7x + 10

2)     Y = 3x² - x – 4

3)     Y = 3x²

4)     Y = x² - 4

EXERCÍCIOS

1) Quais são funções quadráticas?

a)      Y = x² - 5x + 6 (X)
b)      Y = 3x² - 2x + 1 (X)
c)       Y =n 5x – x + 3
d)      Y = 2 + 4x – 1
e)      Y = 5x² (X)
f)       Y = -x² + 4 (X)
g)      Y = 2x – 5
h)      Y= x²/3 – 4x (X)

2) Verifique se a funçao  Y = (2x – 1)² - 4 ( x+1)² é uma função quadratica. (R: não)

3) Obter m na função Y = ( m + 2)x² - 5x + 1 para que seja quadrática
R:  m +  2 ≠ 0 e m≠-2



REPRESENTAÇÃO GRÁFICA DA FUNÇÃO DO 2º GRAU



Vamos atribuir a x valores quaisquer do conjunto dos números reais e calcular o correspondente de y .
O gráfico da função quadrática, quando definida de R em R, é uma curva denominada parábola, como nos seguintes exemplos:

1) Seja a funão definida por:  y = x² -4x + 3
Solução :

Vamos atribuir a x os valores : -1,0,1,2,3,4 e 5 e calcular os valores de y.





















a seguir:

1º) Marcamos os pontos no gráfico.
2º) Traçarmos a curva

















Os ponto V indicado na figura chama-se vértice da parabola


2) Seja a função definida por: Y = -x² -6x - 8

Solução:

Vamos atribuir a x os valores 0,1,2,3,4,5 e 6 e calcular o valor de y
















A seguir
1º) Marcamos os pontos do gráfico.
2º) Traçamos a curva

















O ponto V indicado na figura é o vértice da parábola

3) Seja a função definida por : Y = x² -2x + 1

Solução:

Atribuindo-se valores para x obteremos valores correspondentes para y.

veja:

















A seguir :

1º) Marcamos os pontos no gráfico.
2º) Traçamos a curva





















O ponto V indicado na figura é o vértice da parábola

CARACTERÍSTICAS DA FUNÇÃO QUADRÁTICA

a) Quando à concavidade da parábola:

















b) Quanto às coordenadas do vértice:







EXERCÍCIOS


1) Observe cada função quadrática e responda se o gráfico da parábola tem concavidade para cima ou para baixo,







6) Dada a função y = x²+ 2x+ 2 complete a tabela e esboce o seu gráfico:
a) y= x²-3x + 2
b) y= x² + 4x -3
c) y = x²-4x +4
d) y = -x² + 6x – 9
e) Y = x² - x + 2
f) y = x² - 4
g) y = -x² + 9
h) y = x² - 3x
i) y = x² - 2x – 8
j) y = 2x²


ZEROS DA FUNÇÃO QUADRATICA

Os zeros da função quadrática y = ax² + bx + c,  são os valores de x para os quais y = 0
Então:
Achar os zeros da função quadrática equivale a resolver a equação do 2º grau
ax² + bx + c = 0 ( a diferente de 0)

EXEMPLOS


Dada a função x² - 5x + 6:
a)      Obtenha os zeros da função
b)      Com os zeros obtidos esboce o gráfico da função

Os zeros da função são 2 e 3.



2) Dada a função y = -x² + 6x – 9:


a)      Obtenha os zeros da função.

b)      Com os zeros obtidos esboce o gráfico da função.




3) Dada a função y = x² -2x + 5:

a)      Obtenha os zeros da função.
b)      Com os zeros obtidos esboce o gráfico da função.



EXERCÍCIOS









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