We are given the vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$ The vertex of the function is at $(-5, 1)$, which gives us $h = -5$ and $k = 1$. Therefore, the function so far looks like this: $f(x) = a(x + 5)^2 + 1$

Next, we are given that the quadratic function passes through the point $(-2, 10)$. To find the value of $a$, substitute $x = -2$ and $f(x) = 10$ into the equation: $10 = a(-2 + 5)^2 + 1$ $10 = a(3)^2 + 1$ $10 = 9a + 1$ Now, solve for $a$: $9a = 9 \implies a = 1$

Thus, the quadratic function is: $f(x) = (x + 5)^2 + 1$

Would you like more details or have any further questions?

Here are 5 related questions you might find useful:

1. How do you convert a quadratic equation from vertex form to standard form?
2. What is the significance of the value $a$ in the vertex form of a quadratic equation?
3. How do you find the vertex of a quadratic function given its standard form?
4. How do you determine whether a quadratic function opens upward or downward?
5. Can a quadratic function have more than one vertex?

Tip: In the vertex form of a quadratic function, the value of $h$ represents the x-coordinate of the vertex, and $k$ represents the y-coordinate.