Step 1:
X - intercept are:
$$ \begin{aligned} & \color{blue}{ x_1 = -3-\sqrt{ 3 } } \\[1 em] & \color{blue}{ x_2 = -3+\sqrt{ 3 } } \end{aligned} $$To find the x-intercepts, we need to solve equation $ -x^2-6x-6 = 0 $. (use the quadratic equation solver to view a detailed explanation of how to solve the equation)
Step 2:
Y - intercept is point: $ y-inter=\left(0,~-6\right) $
To find y - coordinate of y - intercept, we need to compute $ f(0) $. In this example we have:
$$ f(\color{blue}{0}) = -1 \cdot \color{blue}{0}^2 -6 \cdot \color{blue}{0} -6 = -6$$Step 3:
Vertex is point: $V=\left(-3,~3\right) $
To find the x - coordinate of the vertex we use formula:
$$ x = -\frac{b}{2a} $$In this example: $ a = -1, b = -6, c = -6 $. So, the x-coordinate of the vertex is:
$$ x = -\frac{b}{2a} = -\frac{ -6 }{ 2 \cdot \left( -1 \right) } = -3 $$$$ y = f \left( -3 \right) = -1 \left( -3 \right)^2 - 6 \cdot \left( -3 \right) ~ - ~ 6 = 3 $$Step 4:
Focus is point: $ F=\left(-3,~\dfrac{ 11 }{ 4 }\right)$
The x - coordinate of the focus is $ x = -\dfrac{b}{2a} $
The y - coordinate of the focus is $ y = \dfrac{1-b^2}{4a} + c $