Step 1:
X - intercept are:
$$ \begin{aligned} & \color{blue}{ x_1 = 7 } \\[1 em] & \color{blue}{ x_2 = 9 } \end{aligned} $$To find the x-intercepts, we need to solve equation $ x^2-16x+63 = 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,~63\right) $
To find y - coordinate of y - intercept, we need to compute $ f(0) $. In this example we have:
$$ f(\color{blue}{0}) = 1 \cdot \color{blue}{0}^2 -16 \cdot \color{blue}{0} + 63 = 63$$Step 3:
Vertex is point: $V=\left(8,~-1\right) $
To find the x - coordinate of the vertex we use formula:
$$ x = -\frac{b}{2a} $$In this example: $ a = 1, b = -16, c = 63 $. So, the x-coordinate of the vertex is:
$$ x = -\frac{b}{2a} = -\frac{ -16 }{ 2 \cdot 1 } = 8 $$$$ y = f \left( 8 \right) = 1 \left( 8 \right)^2 - 16 \cdot 8 ~ + ~ 63 = -1 $$Step 4:
Focus is point: $ F=\left(8,~-\dfrac{ 3 }{ 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 $