Question

para cada função determine os pontos de interseção com os eixos coordenados e i vértice para, em seguida esboçar o grafico a partir desses pontos
$a)h(x) = x {}^{2} + 6x + 8$
$b)f(x) = 5x { }^{2} + 5x$
$c)g(x) = - x {}^{2} + 8x - 12$

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Answer 1

Resposta:

a) x² + 6x + 8

Ponto de intersecção com eixo y: (0,8)

Raízes: x = -2 ou x = -4

Vértice: (-3,-1)

Cálculos:

\begin{gathered}\Delta = 6^2 - 4.1.8\\\\\Delta = 4\\\\\\\sqrt{\Delta} = 2\\\\x_1 = \dfrac{-6+2}{2.1} = -2\\\\x_2 = \dfrac{-6-2}{2.1} = -4\end{gathered}

Δ=6

2

−4.1.8

Δ=4

Δ

=2

x

1

=

2.1

−6+2

=−2

x

2

=

2.1

−6−2

=−4

\begin{gathered}x_v = -\dfrac{6}{2.1} = -3\\\\y_v = -\dfrac{4}{4.1} = -1\end{gathered}

x

v

=−

2.1

6

=−3

y

v

=−

4.1

4

=−1

b) 5x² + 5x

Ponto de intersecção com eixo y: (0,0)

Raízes: x = 0 ou x = -1

Vértice: (-1/2,-5/4)

Cálculos:

5x² + 5x = 5x (x + 1) = 0

x = 0 ou x = -1

\begin{gathered}x_v = -\dfrac{5}{2.5} = -\dfrac12\\\\\\y_v = -\dfrac{25}{4.5} = -\dfrac54\end{gathered}

x

v

=−

2.5

5

=−

2

1

y

v

=−

4.5

25

=−

4

5

c) g(x) = -x² + 8x - 12

Ponto de intersecção com eixo y: (0,-12)

Raízes: x = 2 ou x = 6

Vértice: (4,4)

\begin{gathered}\Delta = 8^2 - 4.(-1).(12)\\\\\Delta = 16\\\\\\\sqrt{\Delta} = 4\\\\x_1 = \dfrac{-8+4}{2.(-1)} = 2\\\\x_2 = \dfrac{-8-4}{2.(-1)} = 6\end{gathered}

Δ=8

2

−4.(−1).(12)

Δ=16

Δ

=4

x

1

=

2.(−1)

−8+4

=2

x

2

=

2.(−1)

−8−4

=6

\begin{gathered}x_v = \dfrac{-8}{2(-1)}= 4\\\\\\y_v = \dfrac{-16}{4(-1)} = 4\end{gathered}

x

v

=

2(−1)

−8

=4

y

v

=

4(−1)

−16

=4