State the parent function. Graph the parent function. Lastly, graph the transformed function and state transformed function and state the transformations. F(x)=-2(x+1)^2+1

Let's break this down step by step:

### 1. **Identify the Parent Function:**
The given function is \( F(x) = -2(x+1)^2 + 1 \).

The parent function is the simplest form of a function type without any transformations. In this case, the parent function is:

\[
f(x) = x^2
\]

This is the basic **quadratic function** (a parabola), which opens upwards and has its vertex at \( (0,0) \).

### 2. **Graph the Parent Function:**
The graph of the parent function \( f(x) = x^2 \) is a parabola with the following properties:
- Vertex: \( (0,0) \)
- Axis of symmetry: \( x = 0 \) (the y-axis)
- Opens upwards
- Symmetrical

### 3. **Transformed Function and its Graph:**
Now, let's analyze the transformed function \( F(x) = -2(x+1)^2 + 1 \). 

#### Steps to describe the transformations:
1. **Horizontal Shift**: 
   The term \( (x + 1)^2 \) means the graph is shifted **1 unit to the left** (opposite of the sign inside the parentheses).

2. **Vertical Stretch and Reflection**: 
   The coefficient \(-2\) causes two changes:
   - The negative sign indicates a **reflection across the x-axis** (the parabola flips upside down).
   - The 2 indicates a **vertical stretch** by a factor of 2, making the parabola narrower.

3. **Vertical Shift**: 
   The \( +1 \) at the end means the graph is shifted **1 unit up**.

#### Final Transformed Function:
\[
F(x) = -2(x+1)^2 + 1
\]

#### Graph Properties:
- **Vertex**: \( (-1, 1) \)
- **Axis of Symmetry**: \( x = -1 \)
- **Direction**: Opens **downwards** (due to the reflection)
- **Narrower Parabola**: Because of the vertical stretch by a factor of 2

### Summary of Transformations:
- **Shift left by 1 unit**
- **Shift up by 1 unit**
- **Reflect across the x-axis**
- **Vertically stretch by a factor of 2**

### Graphing:
I can describe how the graph would look, but I could also generate a visual if you'd like. Would you prefer I generate the graph for the parent function and the transformed function?

---

Do you have any questions or would you like further details? Here are 5 questions to consider:
1. How does a reflection across the x-axis affect the graph of a function?
2. How do you find the vertex of a transformed quadratic function?
3. What effect does a vertical stretch have on a parabola's width?
4. What is the axis of symmetry in a quadratic function and how is it affected by transformations?
5. How would additional transformations (e.g., horizontal stretch) impact the function?

**Tip**: Always remember to identify the vertex first when analyzing quadratic functions—it can give you key insights into transformations.

Explore the transformations of the quadratic function F(x) = -2(x+1)^2 + 1. Learn how the parent function f(x) = x^2 is shifted, stretched, and reflected. This problem covers horizontal and vertical shifts, reflections, and vertical stretches, perfect for students in Grades 9-11.

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