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.

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.