If the absolute value child function g(x)
 has a y
-intercept at (0, 5)
, where is the y
-intercept of g(12x)
? 

The absolute value function \( g(x) \) has a y-intercept at \( (0, 5) \). This means that \( g(0) = 5 \).

Now, we need to find the y-intercept of the transformed function \( g(12x) \). The y-intercept of any function occurs where \( x = 0 \). So, we evaluate \( g(12x) \) at \( x = 0 \):

\[
g(12(0)) = g(0) = 5
\]

Thus, the y-intercept of \( g(12x) \) is still at \( (0, 5) \).

### Key Conclusion:
The y-intercept does not change under horizontal scaling (as in this case), so the y-intercept of \( g(12x) \) remains at \( (0, 5) \).

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What effect does scaling by a factor inside the argument (e.g., \( 12x \)) have on the graph of a function?
2. How does the y-intercept of an absolute value function change if there is a vertical shift?
3. What is the general form of a transformed absolute value function?
4. How would the x-intercept of \( g(x) \) change when transforming to \( g(12x) \)?
5. What would happen to the y-intercept if we scaled vertically instead (e.g., \( 2g(x) \))?

**Tip**: When dealing with transformations of functions, horizontal scaling affects the graph's width but doesn't shift the y-intercept unless there is a vertical translation or reflection.

If the absolute value child function g(x) has a y-intercept at (0, 5), where is the y-intercept of g(12x)?

Explore how the y-intercept of the absolute value function g(x), initially at (0, 5), remains unchanged when the function undergoes horizontal scaling to g(12x). This problem demonstrates key concepts in function transformations and is suitable for Grades 9-11.

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