if the absolute value child function g(x) has a y-intercept at (0,5), where is the y-intercept of g(1/2x)

To find the y-intercept of the function \( g\left(\frac{1}{2}x\right) \), let's break it down step by step.

1. **Understanding the y-intercept of \( g(x) \):**  
   The given information is that the absolute value function \( g(x) \) has a y-intercept at \( (0, 5) \). This means:
   \[
   g(0) = 5
   \]
   
2. **Considering \( g\left(\frac{1}{2}x\right) \):**  
   The function \( g\left(\frac{1}{2}x\right) \) is a horizontal stretch of the function \( g(x) \) by a factor of 2. However, horizontal stretches do not affect the y-intercept because the y-intercept occurs when \( x = 0 \).

3. **Finding the y-intercept of \( g\left(\frac{1}{2}x\right) \):**  
   To find the y-intercept of \( g\left(\frac{1}{2}x\right) \), substitute \( x = 0 \) into the function:
   \[
   g\left(\frac{1}{2} \times 0\right) = g(0)
   \]
   Since we know \( g(0) = 5 \), the y-intercept of \( g\left(\frac{1}{2}x\right) \) is also at \( (0, 5) \).

### Conclusion:
The y-intercept of \( g\left(\frac{1}{2}x\right) \) is the same as the y-intercept of \( g(x) \), which is \( (0, 5) \).

Would you like more details or have any questions? Here are some related questions you might find useful:

1. What happens to the graph of an absolute value function during a horizontal stretch or compression?
2. How does changing the coefficient of \( x \) affect the symmetry of an absolute value function?
3. What are the effects of vertical and horizontal translations on the graph of \( g(x) \)?
4. How can we find the x-intercepts of the function \( g\left(\frac{1}{2}x\right) \)?
5. What are the transformations of the graph \( g(x) = |x| \) for other modifications like \( g\left(2x\right) \)?

**Tip:** Horizontal transformations (stretches or compressions) only affect the x-values, while the y-values (like the y-intercept) stay unchanged unless there's a vertical transformation.

If the absolute value child function g(x) has a y-intercept at (0,5), where is the y-intercept of g(1/2x)?

This detailed explanation covers how to find the y-intercept of the transformed function g(1/2x), given that g(x) has a y-intercept at (0,5). Learn about horizontal stretches and how they affect the graph of absolute value functions. Suitable for students in Grades 9-11.

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