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Let's break down how each graph can be obtained from the graph of \( y = f(x) \):

### a) \( y = -f(x) \)
- This transformation reflects the graph of \( y = f(x) \) across the x-axis. All the y-values of the function are multiplied by -1, flipping the graph upside down.

### b) \( y = f(x + 2) - 3 \)
- The \( (x + 2) \) results in a **horizontal shift** to the left by 2 units.
- The \( -3 \) at the end results in a **vertical shift** down by 3 units.

### c) \( y = -3f(x + 7) \)
- The \( (x + 7) \) results in a **horizontal shift** to the left by 7 units.
- The multiplication by -3 causes two effects:
  1. A **vertical stretch** by a factor of 3 (since the absolute value is greater than 1).
  2. A **reflection** across the x-axis due to the negative sign.

Each part of the transformation adjusts the base graph \( f(x) \) by shifting, stretching, or reflecting it.

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Do you want more detailed explanations for any specific part? 

Here are some related questions to explore:
1. What is the general formula for vertical and horizontal shifts in a function?
2. How do vertical stretches affect the steepness of a graph?
3. How do you identify reflections in graphs based on transformations?
4. What would happen if the function was stretched by a factor of 1/2 instead of 3?
5. How would the graph change if the \( f(x) \) was replaced by \( f(-x) \)?

**Tip:** A vertical shift affects the y-values directly, while a horizontal shift affects the x-values in the opposite direction of the sign.

Describe how the graph of each of the following functions can be obtained from the graph of y = f(x): a) y = -f(x), b) y = f(x + 2) - 3, c) y = -3f(x + 7)

Learn how to apply graph transformations like reflections, horizontal and vertical shifts, and vertical stretches. This problem demonstrates transformations such as y = -f(x), y = f(x + 2) - 3, and y = -3f(x + 7), focusing on changes in position and shape of the graph.

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