List the transformation that occurs to the basic function f(x) for each of the following: y=f(x)+k, y=f(x)-k, y=f(x-h), y=f(x+h), y=-f(x), y=f(-x)

The transformations that occur to the basic function \( f(x) \) for each of the given forms are as follows:

1. **Vertical Translation Upwards**: \( y = f(x) + k \)
   - **Transformation**: The graph of the function is shifted vertically upwards by \( k \) units.
   - **Effect**: Each point on the graph moves \( k \) units higher.

2. **Vertical Translation Downwards**: \( y = f(x) - k \)
   - **Transformation**: The graph of the function is shifted vertically downwards by \( k \) units.
   - **Effect**: Each point on the graph moves \( k \) units lower.

3. **Horizontal Translation to the Right**: \( y = f(x - h) \)
   - **Transformation**: The graph of the function is shifted horizontally to the right by \( h \) units.
   - **Effect**: Each point on the graph moves \( h \) units to the right.

4. **Horizontal Translation to the Left**: \( y = f(x + h) \)
   - **Transformation**: The graph of the function is shifted horizontally to the left by \( h \) units.
   - **Effect**: Each point on the graph moves \( h \) units to the left.

5. **Reflection Across the x-axis**: \( y = -f(x) \)
   - **Transformation**: The graph of the function is reflected across the x-axis.
   - **Effect**: Each point on the graph is inverted vertically; positive \( y \)-values become negative and vice versa.

6. **Reflection Across the y-axis**: \( y = f(-x) \)
   - **Transformation**: The graph of the function is reflected across the y-axis.
   - **Effect**: Each point on the graph is mirrored horizontally; the left side of the graph becomes the right side and vice versa.

Would you like more details on any of these transformations or have any specific questions?

Here are five related questions to expand your understanding:

1. What happens to the function \( y = f(x) \) when it is transformed to \( y = af(x) \)?
2. How does the transformation \( y = f(bx) \) affect the graph of \( f(x) \)?
3. What is the combined effect of \( y = -f(-x) \) on the graph of \( f(x) \)?
4. How does \( y = f(x-h) + k \) transform the graph of \( f(x) \)?
5. What happens to the graph of \( y = f(x) \) when it is transformed to \( y = -f(x-h) + k \)?

**Tip**: When analyzing transformations, it can be helpful to apply each transformation step-by-step to understand their cumulative effect on the graph.

Explore how basic function transformations work, including vertical and horizontal shifts, and reflections. Learn the effects of equations like y=f(x)+k, y=f(x-h), y=-f(x), and y=f(-x) on graph behavior, suitable for students in Grades 9-12.

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