State the parent function. Worked out tea table f(x)=-2(x+1)^2+1

The given function is:

\[
f(x) = -2(x + 1)^2 + 1
\]

### Step-by-Step Breakdown:

1. **Identify the general form**: The function resembles the form of a quadratic equation:
   \[
   f(x) = a(x-h)^2 + k
   \]
   This is the **vertex form** of a parabola, where:
   - \( a \) is the coefficient that affects the width and direction of the parabola.
   - \( h \) is the horizontal shift (the vertex’s x-coordinate).
   - \( k \) is the vertical shift (the vertex’s y-coordinate).

2. **Parent Function**: The **parent function** for any quadratic function is:
   \[
   f(x) = x^2
   \]
   This is a basic parabola centered at the origin \((0, 0)\), which opens upward.

### Comparing \( f(x) = -2(x + 1)^2 + 1 \) to the parent function:
   - The term \( (x + 1) \) indicates a **horizontal shift** of 1 unit to the left (since \( h = -1 \)).
   - The term \( +1 \) outside the squared part indicates a **vertical shift** of 1 unit up (since \( k = 1 \)).
   - The coefficient \( -2 \) in front of the squared term indicates a **vertical stretch** (by a factor of 2) and a **reflection** across the x-axis (since the parabola opens downward).

### Conclusion:
The **parent function** is \( f(x) = x^2 \).

This function has been transformed as follows:
- Shifted 1 unit left.
- Shifted 1 unit up.
- Stretched vertically by a factor of 2.
- Reflected downward.

Do you need more details on any step, or is this clear?

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Here are 5 follow-up questions for further understanding:

1. What is the vertex of the function \( f(x) = -2(x + 1)^2 + 1 \)?
2. How do you determine whether the parabola opens upwards or downwards?
3. What impact does the value of \( a \) have on the shape of the parabola?
4. How do you find the axis of symmetry for a function in vertex form?
5. Can you explain how to rewrite a quadratic function from vertex form to standard form?

**Tip**: When dealing with quadratic functions, the vertex form is especially useful for identifying shifts and the vertex position of the parabola.

Learn how to identify the parent function for the quadratic equation f(x) = -2(x+1)^2 + 1. This problem covers key algebraic concepts, including transformations such as horizontal and vertical shifts, reflection, and vertical stretch. Suitable for students in Grades 9-11.

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