https://file.aiquickdraw.com/mathbot/file/1728756396210k1godixu.jpg

To solve this problem, let's carefully examine the given information:

The equation of the graph is given as \( y = -f(x - 3) \), which indicates two transformations of the function \( f(x) \):

1. **Horizontal Shift**: The \( x - 3 \) part indicates a shift to the right by 3 units.
2. **Reflection**: The negative sign in front of \( f \) means the graph is reflected across the x-axis.

### Steps to solve:

1. **Determine the Slope**: The graph appears to have a positive slope, and we can estimate the slope by choosing two points from the graph. 
   From the graph:
   - The line passes through the points \( (3, 0) \) and \( (0, -7.5) \).
   - The slope is calculated as:
   \[
   \text{slope} = \frac{0 - (-7.5)}{3 - 0} = \frac{7.5}{3} = 2.5 = \frac{5}{2}
   \]
   
2. **Reflect the Slope**: Since the equation has \( -f(x - 3) \), the function is reflected, making the slope \( -\frac{5}{2} \).

3. **Horizontal Shift**: The function is shifted right by 3 units, which means \( f(x - 3) \) must be used instead of just \( f(x) \). So the equation we are looking for should "undo" this shift.

4. **Choose the Correct Answer**: Now, we know the general form of \( f(x) \) should have a slope of \( \frac{5}{2} \) since the reflection has been accounted for by the \( y = -f(x - 3) \) form.

Looking at the given options, the correct equation defining \( f(x) \) is:
\[
f(x) = \frac{5}{2}x + \frac{9}{2}
\]

Thus, the selected answer is correct.

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Do you have any questions or need further details?

Here are 5 related questions to consider:
1. What is the impact of shifting a graph horizontally?
2. How does reflection across the x-axis affect the slope of a line?
3. Can you derive the slope-intercept form of a line given two points?
4. How would a vertical shift affect the equation of the function?
5. What other transformations could affect the graph of a linear function?

**Tip**: When determining the equation of a transformed function, always look for key transformations like translations, reflections, and dilations.

The graph of y = -f(x - 3) is shown. Which equation defines function f?

Learn how to determine the equation of a linear function based on its graph. This example explores graph transformations, including horizontal shifts and reflections, to find the correct slope and intercept. Suitable for students in Grades 10-12.

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