To find the vertex of the parabola given by the equation:
\[ y = x^2 - 4x + \frac{46}{5}, \]
we can use the vertex formula. The vertex \( (h, k) \) of a parabola given by the standard form \( y = ax^2 + bx + c \) can be found using the formula:
- \( h = -\frac{b}{2a} \)
- \( k = f(h) \) (substituting \( h \) back into the equation to find \( k \))
In your case, we have:
- \( a = 1 \)
- \( b = -4 \)
- \( c = \frac{46}{5} \)
Now, calculate \( h \):
\[ h = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2. \]
Next, substitute \( h \) back into the equation to find \( k \):
\[ k = f(2) = (2)^2 - 4(2) + \frac{46}{5}. \]
Calculating this step by step:
- \( (2)^2 = 4 \)
- \( -4(2) = -8 \)
- Combine the terms: \[ k = 4 - 8 + \frac{46}{5} = -4 + \frac{46}{5}. \]
Now convert \(-4\) to a fraction with a denominator of \(5\):
\[ -4 = \frac{-20}{5}. \]
Now combine the fractions:
\[ k = \frac{-20}{5} + \frac{46}{5} = \frac{26}{5}. \]
Thus, the vertex \( (h, k) \) is:
\[ \left( 2, \frac{26}{5} \right). \]
So, the vertex of the parabola is:
\[ \boxed{\left( 2, \frac{26}{5} \right)}. \]