To find the \(a\)-value and \(c\)-value (the y-intercept) of the quadratic function represented by a parabola (specifically in vertex form), we can use the general form of a parabola:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In this case, you have the vertex \((-3, -10)\), so \(h = -3\) and \(k = -10\).
Thus, the equation can be expressed as:
\[ y = a(x + 3)^2 - 10 \]
You mentioned that the graph crosses the y-axis at \(-1\). This means when \(x = 0\), \(y = -1\). We can use this point to find the value of \(a\).
Substituting \(x = 0\) and \(y = -1\) into the equation:
\[ -1 = a(0 + 3)^2 - 10 \]
This simplifies to:
\[ -1 = a(3)^2 - 10 \] \[ -1 = 9a - 10 \] \[ 9a = -1 + 10 \] \[ 9a = 9 \] \[ a = 1 \]
Now, since the y-intercept \((c)\) is the value of \(y\) when \(x = 0\):
\[ c = -1 \]
To summarize:
- The \(a\)-value is \(1\).
- The \(c\)-value (y-intercept) is \(-1\).
The final equation of the parabola is:
\[ y = (x + 3)^2 - 10 \]