To write a quadratic function in vertex form given the vertex and a point the function passes through, we can follow these steps:
1. **Find the vertex form of a quadratic function:**
The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where (h, k) is the vertex of the parabola.
2. **Use the given vertex and point to find the value of 'a':**
Given vertex (-2, -6) and point (-1, 2), we can substitute these values into the vertex form to solve for 'a'.
3. **Substitute the vertex into the vertex form:**
Using the vertex (-2, -6):
\[ f(x) = a(x - (-2))^2 - 6 \]
\[ f(x) = a(x + 2)^2 - 6 \]
4. **Substitute the point into the equation and solve for 'a':**
Using the point (-1, 2):
\[ 2 = a(-1 + 2)^2 - 6 \]
\[ 2 = a(1)^2 - 6 \]
\[ 2 = a - 6 \]
\[ a = 2 + 6 \]
\[ a = 8 \]
5. **Write the quadratic function with the found 'a' value:**
Now that we have found 'a' to be 8, we can write the quadratic function in vertex form:
\[ f(x) = 8(x + 2)^2 - 6 \]
Therefore, the quadratic function in vertex form that satisfies the given conditions is:
\[ f(x) = 8(x + 2)^2 - 6 \]
