To find the equation of a quadratic function in standard form \(y = ax^2 + bx + c\) that passes through the point \((3, -1)\), we need more information or constraints about the quadratic. However, if we assume that this point is the vertex of the quadratic, we can use the vertex form of the quadratic equation:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex. Here, \((h, k) = (3, -1)\).
Thus, the equation becomes:
\[ y = a(x - 3)^2 - 1 \]
To convert this to standard form, we need to know the value of \(a\). If we assume a specific value for \(a\), we can expand the equation accordingly. However, let's say we choose \(a = 1\) for simplicity:
\[ y = (x - 3)^2 - 1 \]
Now, we can expand this:
\[ y = (x^2 - 6x + 9) - 1 \] \[ y = x^2 - 6x + 8 \]
So the standard form of the quadratic function, assuming the vertex at (3, -1) and \(a = 1\), is:
\[ y = 1x^2 - 6x + 8 \]
Therefore, in the blanks you would fill in:
Standard form of the quadratic is: \(y = 1x^2 - 6x + 8\).