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To solve this problem, we need to find the values of \( a \), \( h \), and \( k \) for the absolute value function given in vertex form:

\[
f(x) = a |x - h| + k
\]

where:
- The vertex of the function is given as \( (h, k) = (7, -3) \).
- The function passes through the point \( (3, -9) \).

### Step 1: Substitute \( h \) and \( k \)
From the vertex, we know:
\[
h = 7 \quad \text{and} \quad k = -3
\]

So, the function becomes:
\[
f(x) = a |x - 7| - 3
\]

### Step 2: Use the point \( (3, -9) \) to find \( a \)
Since the function passes through \( (3, -9) \), we substitute \( x = 3 \) and \( f(x) = -9 \) into the equation to solve for \( a \):

\[
-9 = a |3 - 7| - 3
\]

Calculate \( |3 - 7| \):
\[
|3 - 7| = |-4| = 4
\]

Now, substitute this into the equation:
\[
-9 = a \cdot 4 - 3
\]

Add 3 to both sides:
\[
-6 = 4a
\]

Divide by 4:
\[
a = -\frac{3}{2}
\]

### Final Answer
The values are:
- \( a = -\frac{3}{2} \)
- \( h = 7 \)
- \( k = -3 \)

Thus, the equation of the function is:
\[
f(x) = -\frac{3}{2} |x - 7| - 3
\]

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Would you like further explanation on any of these steps?

### Related Questions
1. How do you determine the slope of a line passing through two points?
2. What is the significance of the vertex in an absolute value function?
3. How would the graph of this function look based on the values of \( a \), \( h \), and \( k \)?
4. How would this function change if the point was different but the vertex stayed the same?
5. How do you find the y-intercept of an absolute value function?

### Tip
Remember, in absolute value functions, the vertex represents the "turning point," where the graph changes direction.

An absolute value function has its vertex at the point (7, −3). The function passes through the point (3, −9). When written in vertex form, the function is f(x) = a|x − h| + k, where: a = ?, h = ?, k = ?. No decimal answer. The equation is ____________.

Learn how to find the vertex form of an absolute value function with a given vertex and a point. This problem explores the properties of absolute value functions and demonstrates how to determine the parameters a, h, and k. Suitable for high school students in Grades 9-12.

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