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The diagram shows the curve \( y = f(x) \), with its minimum point at \( (2, -1) \). You are asked to find the turning points (minimum or maximum) of new functions derived from transformations of \( f(x) \).

To solve this:

### 1. Transformation rules:
- \( y = f(x + a) \): This translates the graph horizontally by \( -a \). Positive \( a \) shifts the graph left by \( a \) units, and negative \( a \) shifts it right.
- \( y = f(x) + a \): This translates the graph vertically by \( a \). Positive \( a \) shifts it up, and negative \( a \) shifts it down.
- \( y = -f(x) \): Reflects the graph vertically (about the x-axis).
- \( y = f(-x) \): Reflects the graph horizontally (about the y-axis).

The minimum point of the original curve is \( (2, -1) \). Let's apply these transformations one by one.

### Answers:
a) \( y = f(x + 2) \):
- Horizontal shift left by 2. The new minimum point is at \( (0, -1) \).

b) \( y = f(x - 2) \):
- Horizontal shift right by 2. The new minimum point is at \( (4, -1) \).

c) \( y = f(x) + 2 \):
- Vertical shift up by 2. The new minimum point is at \( (2, 1) \).

d) \( y = f(x) - 2 \):
- Vertical shift down by 2. The new minimum point is at \( (2, -3) \).

e) \( y = -f(x) \):
- Reflection in the x-axis. The new point is a maximum at \( (2, 1) \).

f) \( y = -f(x) + 2 \):
- Reflection in the x-axis and shift up by 2. The new point is a maximum at \( (2, 3) \).

Would you like further details or explanations on any of the transformations?

Here are 5 related questions:
1. How does the transformation \( y = f(x + a) \) affect the curve's behavior?
2. What is the result of reflecting a curve around the x-axis?
3. Can a curve have more than one turning point?
4. How do vertical shifts affect the curvature of the graph?
5. What happens when both horizontal and vertical transformations are applied simultaneously?

**Tip**: Always remember that horizontal shifts affect the x-coordinate, while vertical shifts impact the y-coordinate.

Write down the coordinates of the turning point of the curve with equation: a) y = f(x + 2), b) y = f(x - 2), c) y = f(x) + 2, d) y = f(x) - 2, e) y = -f(x), f) y = -f(x) + 2.

Explore the turning points of transformed functions including horizontal shifts, vertical shifts, and reflections. This page covers transformation rules for functions, key examples, and visual diagrams to enhance understanding. Suitable for high school students studying advanced algebra.

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