Graph $ \color{blue}{ f(x) = 2x^2-20x+46} $

Step 1:

X - intercept are:

$$ \begin{aligned} & \color{blue}{ x_1 = 5-\sqrt{ 2 } } \\[1 em] & \color{blue}{ x_2 = 5+\sqrt{ 2 } } \end{aligned} $$

To find the x-intercepts, we need to solve equation $ 2x^2-20x+46 = 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,~46\right) $

To find y - coordinate of y - intercept, we need to compute $ f(0) $. In this example we have:

$$ f(\color{blue}{0}) = 2 \cdot \color{blue}{0}^2 -20 \cdot \color{blue}{0} + 46 = 46$$

Step 3:

Vertex is point: $V=\left(5,~-4\right) $

To find the x - coordinate of the vertex we use formula:

$$ x = -\frac{b}{2a} $$

In this example: $ a = 2, b = -20, c = 46 $. So, the x-coordinate of the vertex is:

$$ x = -\frac{b}{2a} = -\frac{ -20 }{ 2 \cdot 2 } = 5 $$$$ y = f \left( 5 \right) = 2 \left( 5 \right)^2 - 20 \cdot 5 ~ + ~ 46 = -4 $$

Step 4:

Focus is point: $ F=\left(5,~-\dfrac{ 31 }{ 8 }\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 $

Quadratic Function Grapher