16.	The vertex of the graph of a quadratic function y = f(x) is (–4 , 7). If the graph passes through the point (–1 , –2), find f(x).

Given that the vertex of the quadratic function \( y = f(x) \) is \((-4, 7)\), we can express the quadratic function in the vertex form:

\[ f(x) = a(x + 4)^2 + 7 \]

We also know that the function passes through the point \((-1, -2)\). We can use this information to find the value of \(a\).

Substitute \((-1, -2)\) into the equation:

\[ -2 = a(-1 + 4)^2 + 7 \]

Simplify inside the parentheses:

\[ -2 = a(3)^2 + 7 \]

\[ -2 = 9a + 7 \]

Solve for \(a\):

\[ -2 - 7 = 9a \]

\[ -9 = 9a \]

\[ a = -1 \]

Now substitute \(a\) back into the equation:

\[ f(x) = -1(x + 4)^2 + 7 \]

Thus, the quadratic function is:

\[ f(x) = -(x + 4)^2 + 7 \]

Would you like more details or have any questions?

Here are some additional questions you might find useful:

1. How do you convert a quadratic function from standard form to vertex form?
2. What is the significance of the vertex in a quadratic function?
3. How do you determine the axis of symmetry for a quadratic function?
4. What are the steps to find the roots of a quadratic equation?
5. How does changing the coefficient \(a\) in a quadratic function affect its graph?

**Tip:** Always double-check your work by substituting known points back into the equation to verify your solution.

Learn how to find the quadratic function f(x) = -(x + 4)^2 + 7 passing through the point (–1, –2) with its vertex at (–4, 7). Understand the vertex form of quadratic functions and how to use substitution to determine function coefficients. Suitable for Grades 9-12.

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