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Part 1
Explain why the graph of the equation g left parenthesis x right parenthesis equals negative left parenthesis x plus 1 right parenthesis squared minus 3 would be a parabola opening downward.
Question content area bottom
Part 1
The parabola opens downwards since aequals

enter your response here and this value is
▼
equal to
less than
greater than
0.
​(Type an integer or a​ decimal.)
The parabola opens downwards since aand this value isand this value is(Type an integer or a decimal.)
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The equation given is \( g(x) = - (x + 1)^2 - 3 \). 

### Explanation
This equation represents a quadratic function in the form \( g(x) = a(x - h)^2 + k \), where:
- \( a \) determines the direction of the parabola (upwards or downwards).
- \( (h, k) \) represents the vertex of the parabola.

For the given equation:
- The coefficient \( a = -1 \).
- The expression \( (x + 1)^2 \) indicates a horizontal shift of the graph.
- The constant term \( -3 \) indicates a vertical shift downwards.

Since \( a = -1 \) (which is less than 0), the parabola opens downward. This is because the negative sign in front of the squared term causes the parabola to flip vertically.

### Final Answer
The parabola opens downwards since \( a = -1 \) and this value is less than 0.

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Would you like more details or have any further questions? Here are some related questions you might explore:

1. What is the vertex of the parabola described by this equation?
2. How would the graph change if the equation were \( g(x) = (x + 1)^2 - 3 \)?
3. What is the axis of symmetry for the graph of this function?
4. How do we determine the direction in which any quadratic equation opens?
5. How does the value of \( a \) affect the width of the parabola?

**Tip:** The value of \( a \) in a quadratic equation not only determines the direction of the parabola but also its "steepness." The larger the absolute value of \( a \), the narrower the parabola.

Understand why the graph of the equation g(x) = -(x + 1)^2 - 3 opens downward. Learn about quadratic functions and the vertex form of a parabola. Suitable for high school students.

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