https://file.aiquickdraw.com/mathbot/file/1728436388477sq3sknq8.jpg

The function provided is \( f(x) = 3 \lceil x \rceil + 9 \), where \( \lceil x \rceil \) represents the *ceiling function*, which rounds up \( x \) to the nearest integer greater than or equal to \( x \). This results in a step function that moves in discrete jumps.

### Understanding the graph of \( f(x) = 3 \lceil x \rceil + 9 \):

1. The graph will have discrete points for each integer value of \( x \).
2. At each integer \( x \), the value of \( f(x) \) increases by a jump of 3 units because of the factor \( 3 \).
3. The overall vertical shift of the graph is by 9 units, so all steps begin from 9.

Let's break it down for various values of \( x \):

- When \( x = 0 \), \( f(x) = 3(0) + 9 = 9 \).
- When \( x = 1 \), \( f(x) = 3(1) + 9 = 12 \).
- When \( x = 2 \), \( f(x) = 3(2) + 9 = 15 \).
- And so on, increasing by 3 for each unit change in \( x \).
- On the negative side, the graph will have a similar pattern: for example, \( x = -1 \) gives \( f(x) = 3(-1) + 9 = 6 \).

### Analyzing the Graph Choices:

From the four choices in the image:

- **Graph a**: This graph appears to show steps in the correct direction but is shifted too far right and up. 
- **Graph b**: This shows incorrect behavior for negative values and is not a correct match.
- **Graph c**: This seems to represent a correct step function with jumps of 3 units and a vertical shift starting at 9.
- **Graph d**: Incorrect as the steps aren't consistent with the expected pattern.

### Conclusion:
The correct graph that matches the function \( f(x) = 3 \lceil x \rceil + 9 \) is **Graph c**.

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Do you need further details or have any questions?

Here are 5 questions to deepen your understanding:
1. How does the ceiling function \( \lceil x \rceil \) impact the shape of the graph?
2. What would the graph look like if it used the floor function \( \lfloor x \rfloor \) instead?
3. How would the graph change if the vertical shift was different, say \( f(x) = 3 \lceil x \rceil + 5 \)?
4. Can you explain why the graph increases by steps of 3?
5. How would the graph behave for non-integer values of \( x \)?

**Tip**: When graphing step functions, always pay attention to how rounding (floor or ceiling) affects the function's behavior at integer and non-integer values.

Explore the graph of the step function f(x) = 3⌈x⌉ + 9, where ⌈x⌉ is the ceiling function. Learn how to analyze its jumps, vertical shifts, and overall behavior. Suitable for high school math students in Grades 9-12.

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