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The problem asks you to graph the function \( f(x) = (x + 2)^2 - 1 \) using transformations starting from the graph of \( y = x^2 \), the basic parabola. Let's break it down step by step:

### 1. **Base Function:**
   The base function is \( y = x^2 \), which is a standard parabola opening upwards with its vertex at the origin \( (0, 0) \).

### 2. **Transformation 1: Horizontal Shift**
   - The term \( (x + 2) \) indicates a horizontal shift.
   - Since the transformation involves \( +2 \), the graph shifts **2 units to the left**.
   - The new vertex is now at \( (-2, 0) \).

### 3. **Transformation 2: Vertical Shift**
   - The term \( -1 \) represents a vertical shift.
   - This means the entire graph is shifted **1 unit down**.
   - The new vertex after this transformation is at \( (-2, -1) \).

### 4. **Summary of Transformations:**
   - Start with the graph of \( y = x^2 \) (a parabola).
   - Shift the graph 2 units to the left.
   - Then, shift the graph 1 unit downward.

The final function \( f(x) = (x + 2)^2 - 1 \) is a parabola with its vertex at \( (-2, -1) \), opening upwards.

Would you like me to graph this function for you?

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### Related Questions:
1. How do vertical shifts affect the graph of a function?
2. What is the effect of horizontal shifts on a parabola?
3. How do you determine whether a transformation involves stretching or compressing?
4. Can you describe how a reflection would affect the function \( y = x^2 \)?
5. How would the graph change if we had \( f(x) = -(x + 2)^2 - 1 \)?

#### Tip: 
Transformations on functions follow the order: first, handle horizontal changes (inside the parentheses), then vertical changes (outside).

Graph the function f(x) = (x + 2)^2 - 1 by starting with the graph of y = x^2 and using transformations (shifting, stretching/compressing, and/or reflecting).

Learn how to graph the quadratic function f(x) = (x + 2)^2 - 1 by applying transformations to the base graph of y = x^2. This guide covers horizontal and vertical shifts, providing a step-by-step solution suitable for high school students in Grades 9-12.

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