To write a quadratic function in the vertex form $f(x) = a(x - h)^2 + k$, we need to find the value of $a$, given that the vertex $(h, k)$ is $(8, -9)$ and the graph passes through the point $(5, -18)$.

Step-by-Step Solution:

1. Identify the given information:

   • Vertex $(h, k) = (8, -9)$
   • Point on the graph $(x, f(x)) = (5, -18)$

2. Substitute the vertex into the vertex form: $f(x) = a(x - 8)^2 - 9$

3. Use the point $(5, -18)$ to find $a$: Substitute $x = 5$ and $f(x) = -18$ into the equation: $-18 = a(5 - 8)^2 - 9$

4. Simplify the equation: $-18 = a(-3)^2 - 9$ $-18 = a(9) - 9$

5. Solve for $a$: $-18 + 9 = 9a$ $-9 = 9a$ $a = -1$

6. Write the quadratic function in vertex form: $f(x) = -1(x - 8)^2 - 9$ or $f(x) = -(x - 8)^2 - 9$

Conclusion:

The quadratic function in vertex form that opens down, has a vertex at $(8, -9)$, and passes through the point $(5, -18)$ is:

$f(x) = -(x - 8)^2 - 9$

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Related Questions:

1. How do you convert a quadratic function from vertex form to standard form?
2. What are the characteristics of a parabola that opens downward?
3. How do you find the axis of symmetry for a quadratic function in vertex form?
4. How can you find the roots of a quadratic function given in vertex form?
5. How does changing the value of $a$ in the vertex form affect the graph of the parabola?

Tip:

For a quadratic function in vertex form, the sign of $a$ determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$).