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?

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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.