### Instruction: Find the focus of the parabola $y = \frac{5x^2 - 4x + 2}{4}.$.### Response: I know that a parabola of the form $y = ax^2 + bx + c$ has a focus at the point $(h, k + \frac{1}{4a}),$ where $h = -\frac{b}{2a}$ and $k = -\frac{b^2 - 4ac}{4a}.$
So I want to find the values of $a, b, c, h, k$ for this parabola.
I see that $a = \frac{5}{4}, b = -1, c = \frac{1}{2}$ by comparing the given equation with the general form.
To find $h,$ I plug in $a$ and $b$ into the formula $h = -\frac{b}{2a}.$
I get $h = -\frac{-1}{2 \cdot \frac{5}{4}} = \frac{2}{5}.$
To find $k,$ I plug in $a, b, c$ into the formula $k = -\frac{b^2 - 4ac}{4a}.$
I get $k = -\frac{(-1)^2 - 4 \cdot \frac{5}{4} \cdot \frac{1}{2}}{4 \cdot \frac{5}{4}} = -\frac{1 - 5}{5} = \frac{4}{5}.$
To find the focus, I add $\frac{1}{4a}$ to the $y$-coordinate of $k.$
I get $\frac{4}{5} + \frac{1}{4a} = \frac{4}{5} + \frac{1}{4 \cdot \frac{5}{4}} = \frac{4}{5} + \frac{1}{5} = 1.$
So the focus of the parabola is $(\frac{2}{5}, 1).$