Question

On a graph the vertex is (-3,-10) and the y intercept crosses at -1
The graph is opens upwards looking for a-value and c-value

Answer 1

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:

\[
y = (x + 3)^2 - 10
\]