Graph $ \color{blue}{ f(x) = x^2+\dfrac{173}{100}x+\dfrac{433}{500}} $
Step 1:
x - intercept does not exist.
To find the x-intercepts, we need to solve equation $ x^2+\dfrac{173}{100}x+\dfrac{433}{500} = 0 $. Since equation $ x^2+\dfrac{173}{100}x+\dfrac{433}{500} = 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{ 433 }{ 500 }\right) $
To find y - coordinate of y - intercept, we need to compute $ f(0) $. In this example we have:
$$ f(\color{blue}{0}) = 1 \cdot \color{blue}{0}^2 + \frac{ 173 }{ 100 } \cdot \color{blue}{0} + \frac{ 433 }{ 500 } = \frac{ 433 }{ 500 }$$Step 3:
Vertex is point: $V=\left(-\dfrac{ 173 }{ 200 },~\dfrac{ 4711 }{ 40000 }\right) $
To find the x - coordinate of the vertex we use formula:
$$ x = -\frac{b}{2a} $$In this example: $ a = 1, b = \frac{ 173 }{ 100 }, c = \frac{ 433 }{ 500 } $. So, the x-coordinate of the vertex is:
$$ x = -\frac{b}{2a} = -\frac{ \frac{ 173 }{ 100 } }{ 2 \cdot 1 } = -\frac{ 173 }{ 200 } $$$$ y = f \left( -\frac{ 173 }{ 200 } \right) = 1 \left( -\frac{ 173 }{ 200 } \right)^2 + \frac{ 173 }{ 100 } \cdot \left( -\frac{ 173 }{ 200 } \right) ~ + ~ \frac{ 433 }{ 500 } = \frac{ 4711 }{ 40000 } $$Step 4:
Focus is point: $ F=\left(-\dfrac{ 173 }{ 200 },~\dfrac{ 14711 }{ 40000 }\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 $