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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.

Write a quadratic function in the vertex form f(x) = a(x − h)^2 + k such that the graph of the function opens up and has a vertex at (−5,1) and contains the point (−2,10).

Learn how to write a quadratic function in vertex form f(x) = a(x − h)^2 + k with a vertex at (−5,1) and a point at (−2,10). This solution explains step-by-step how to find the value of 'a' and construct the full equation.

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