how does the graph of y=f(x) relate to the graph of y=f(abs(x))?

The relationship between the graphs of \( y = f(x) \) and \( y = f(|x|) \) is as follows:

1. **Right Half ( \( x \geq 0 \) ):**
   - For \( x \geq 0 \), \( f(|x|) = f(x) \) because the absolute value of a positive number or zero is the number itself. Hence, the graph of \( y = f(|x|) \) matches exactly with the graph of \( y = f(x) \) for \( x \geq 0 \).

2. **Left Half ( \( x < 0 \) ):**
   - For \( x < 0 \), \( f(|x|) = f(-x) \) because the absolute value of a negative number is its positive counterpart. So, the graph of \( y = f(|x|) \) on the left side of the y-axis (where \( x < 0 \)) is the **reflection of the right half** of the graph of \( y = f(x) \) about the y-axis.

### Summary:
- The graph of \( y = f(|x|) \) is obtained by **reflecting** the portion of the graph of \( y = f(x) \) for \( x \geq 0 \) (the right half) over the y-axis, and ignoring the original left half of \( y = f(x) \).

### Example:
- If \( f(x) = x^2 \), the graph of \( y = f(x) \) is a parabola that opens upward with its vertex at the origin.
- The graph of \( y = f(|x|) = |x|^2 = x^2 \) would look identical, since squaring makes everything positive, and there is no visible difference between the two graphs.
  
For other functions, this reflection effect becomes clearer.

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

1. How does the graph of \( y = f(x) \) compare to \( y = |f(x)| \)?
2. What happens if you consider \( y = f(x - h) \) instead of \( y = f(|x|) \)?
3. How would the graph change if the function were piecewise or discontinuous?
4. Can we generalize the behavior for specific types of functions like exponential or logarithmic?
5. How does this reflection affect the domain and range of the function?

**Tip**: When analyzing transformations involving absolute values, always consider how the graph behaves for both positive and negative inputs!

How does the graph of y=f(x) relate to the graph of y=f(abs(x))?

Explore how the graph of y=f(x) relates to y=f(abs(x)) with step-by-step explanations. Learn about the reflection across the y-axis and how absolute value functions affect the shape of graphs. Suitable for high school students in Grades 9-12.

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