Graph $ \color{blue}{ f(x) = \dfrac{123}{500}x^2-\dfrac{567}{250}x+\dfrac{7309}{1000}} $
Step 1:
x - intercept does not exist.
To find the x-intercepts, we need to solve equation $ \dfrac{123}{500}x^2-\dfrac{567}{250}x+\dfrac{7309}{1000} = 0 $. Since equation $ \dfrac{123}{500}x^2-\dfrac{567}{250}x+\dfrac{7309}{1000} = 0 $ does not have real soultions we conclude that the function does not have x - intercepts too. (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,~\dfrac{ 7309 }{ 1000 }\right) $
To find y - coordinate of y - intercept, we need to compute $ f(0) $. In this example we have:
$$ f(\color{blue}{0}) = \frac{ 123 }{ 500 } \cdot \color{blue}{0}^2 -\frac{ 567 }{ 250 } \cdot \color{blue}{0} + \frac{ 7309 }{ 1000 } = \frac{ 7309 }{ 1000 }$$Step 3:
Vertex is point: $V=\left(\dfrac{ 189 }{ 41 },~\dfrac{ 85343 }{ 41000 }\right) $
To find the x - coordinate of the vertex we use formula:
$$ x = -\frac{b}{2a} $$In this example: $ a = \frac{ 123 }{ 500 }, b = -\frac{ 567 }{ 250 }, c = \frac{ 7309 }{ 1000 } $. So, the x-coordinate of the vertex is:
$$ x = -\frac{b}{2a} = -\frac{ -\frac{ 567 }{ 250 } }{ 2 \cdot \frac{ 123 }{ 500 } } = \frac{ 189 }{ 41 } $$$$ y = f \left( \frac{ 189 }{ 41 } \right) = \frac{ 123 }{ 500 } \left( \frac{ 189 }{ 41 } \right)^2 - \frac{ 567 }{ 250 } \cdot \frac{ 189 }{ 41 } ~ + ~ \frac{ 7309 }{ 1000 } = \frac{ 85343 }{ 41000 } $$Step 4:
Focus is point: $ F=\left(\dfrac{ 189 }{ 41 },~\dfrac{ 381029 }{ 123000 }\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 $