To convert the graph of $f(x) = 3|x| + 3$ into the graph of $g(x) = 9|x| + 3$, the transformation involves a vertical stretch.

Explanation:

1. The structure of both functions is $a|x| + b$, where $a$ determines the steepness or vertical stretch/compression of the graph, and $b$ represents the vertical shift.
2. Comparing $f(x) = 3|x| + 3$ and $g(x) = 9|x| + 3$:
   • The constant $b = 3$ is the same in both equations, so there is no vertical shift.
   • The coefficient $a$ changes from 3 to 9, indicating the graph becomes steeper by a factor of 3.

Transformation:

• A vertical stretch by a factor of 3 is applied to the graph of $f(x)$.

Would you like a detailed visual or further breakdown?

---

Here are 5 related questions to expand your understanding:

1. What transformation would change $f(x) = 3|x| + 3$ into $h(x) = 3|x| - 2$?
2. How does the graph of $f(x) = |x|$ compare to $f(x) = a|x| + b$ in general?
3. If the coefficient of $|x|$ is negative, how does the graph change?
4. How would a horizontal stretch or compression affect the graph of $|x|$?
5. Can you describe the effect of changing both $a$ and $b$ simultaneously in $a|x| + b$?

Tip:

Always compare the coefficients of $|x|$ and the constants separately to identify the transformations applied to the graph!