What does the transformation f(x)↦f(x+3)+10 do to the graph of f(x)?

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The transformation \( f(x) \mapsto f(x+3) + 10 \) applies two changes to the graph of \( f(x) \):

1. **Horizontal Shift**: The term \( f(x+3) \) represents a horizontal shift to the left by 3 units. This means every point on the graph of \( f(x) \) moves 3 units to the left.

2. **Vertical Shift**: The term \( +10 \) indicates a vertical shift upwards by 10 units. This means that after the horizontal shift, the entire graph moves 10 units upward.

### Combined Effect:
The graph of \( f(x) \) is shifted 3 units to the left and then 10 units upward.

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Do you want details or have any questions? Here are some related questions:

1. What happens if the transformation is \( f(x) \mapsto f(x-5) - 7 \)?
2. How would the graph change if the transformation was \( f(x) \mapsto -f(x+3) + 10 \)?
3. What does the transformation \( f(x) \mapsto 2f(x) \) do to the graph of \( f(x) \)?
4. How does the transformation \( f(x) \mapsto f(-x) \) affect the graph?
5. What happens if you apply the transformation \( f(x) \mapsto f(2x) \)?
6. How would the graph change with \( f(x) \mapsto f(x) + c \) where \( c \) is a constant?
7. What does the transformation \( f(x) \mapsto f(x) - 3 \) do to the graph?
8. What effect does the transformation \( f(x) \mapsto f(0.5x) \) have on the graph?

**Tip**: For horizontal shifts, \( f(x + c) \) shifts the graph left if \( c > 0 \) and right if \( c < 0 \).

Explore how the transformation f(x)↦f(x+3)+10 affects the graph of a function. Learn about horizontal and vertical shifts in graphs, with detailed explanations and examples. Suitable for high school students studying functions and transformations.

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