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Part 1
Use transformations of f left parenthesis x right parenthesis equals x squared to graph the following function.
g left parenthesis x right parenthesis equals negative 2 left parenthesis x plus 3 right parenthesis squared minus 2
Question content area bottom left
Part 1
Select all the transformations that are needed to graph the given function using f left parenthesis x right parenthesis equals x squared.
A.
Shift the graph 3 units to the left.
B.
Reflect the graph about the​ x-axis.
C.
Shift the graph 2 units down.
D.
Shift the graph 2 units up.
E.
Stretch the graph vertically by a factor of 2.
F.
Shrink the graph vertically by a factor of 2.
G.
Reflect the graph about the​ y-axis.
H.
Stretch the graph horizontally by a factor of 2.
I.
Shift the graph 3 units to the right.
J.
Shrink the graph horizontally by a factor of 2.
.
.
.
Select all the transformations that are needed to graph the given function using f left parenthesis x right parenthesis equals x squared.
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To solve this problem, we need to determine the transformations that are applied to the base function \( f(x) = x^2 \) to obtain the new function \( g(x) = -2(x + 3)^2 - 2 \).

### Step-by-Step Analysis:

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

2. **Transformation to \( g(x) = -2(x + 3)^2 - 2 \)**:
   - **Horizontal Shift**:
     - The expression inside the square, \( (x + 3) \), indicates a shift.
     - \( x + 3 \) implies a **shift 3 units to the left** (opposite to the sign inside the parentheses).
     - **Corresponding Transformation**: \( A \) (Shift the graph 3 units to the left).

   - **Reflection**:
     - The negative sign in front of the expression \( -2(x + 3)^2 \) indicates a **reflection about the x-axis**.
     - **Corresponding Transformation**: \( B \) (Reflect the graph about the x-axis).

   - **Vertical Stretch**:
     - The factor of \( -2 \) outside the square implies a **vertical stretch by a factor of 2** (the negative sign is for reflection, and the absolute value 2 is the stretch factor).
     - **Corresponding Transformation**: \( E \) (Stretch the graph vertically by a factor of 2).

   - **Vertical Shift**:
     - The \(-2\) at the end of the function \( -2(x + 3)^2 - 2 \) indicates a **shift 2 units down**.
     - **Corresponding Transformation**: \( C \) (Shift the graph 2 units down).

### Conclusion:
The transformations needed to graph \( g(x) = -2(x + 3)^2 - 2 \) from \( f(x) = x^2 \) are:
- **A**: Shift the graph 3 units to the left.
- **B**: Reflect the graph about the x-axis.
- **C**: Shift the graph 2 units down.
- **E**: Stretch the graph vertically by a factor of 2.

These are the correct transformations.

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Would you like more details on how these transformations affect the graph? 

### Related Questions:
1. What would happen if the function was \( g(x) = 2(x + 3)^2 + 2 \)?
2. How would the graph change if the function was \( g(x) = -\frac{1}{2}(x - 3)^2 - 2 \)?
3. What is the vertex of the function \( g(x) = -2(x + 3)^2 - 2 \)?
4. How do horizontal and vertical stretches affect the graph of a quadratic function?
5. What is the difference between reflecting a graph about the x-axis and the y-axis?

**Tip**: When analyzing transformations, always apply the horizontal shifts and reflections first, followed by vertical transformations.

Learn how to graph the function g(x) = -2(x + 3)^2 - 2 by applying transformations to the base function f(x) = x^2. Understand horizontal shifts, reflections, vertical stretches, and shifts. Suitable for high school students studying quadratic functions.

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