https://file.aiquickdraw.com/mathbot/file/17295242233700jmlq54b.jpg

The problem asks to determine the direction of the opening of each quadratic function from the image. For quadratic functions, the key to determining whether the graph opens upwards or downwards is based on the coefficient of the \(x^2\) term. Specifically:

- If the coefficient of \(x^2\) is **positive**, the parabola opens **upward**.
- If the coefficient of \(x^2\) is **negative**, the parabola opens **downward**.

Let’s analyze each equation in the image:

1. **\( y = (x + 5)^2 - 2 \)**
   - Expand the equation: \( y = (x + 5)^2 - 2 = x^2 + 10x + 25 - 2 = x^2 + 10x + 23 \).
   - The coefficient of \(x^2\) is **positive** \( (1) \), so the parabola opens **upward**.

2. **\( y = -2x^2 + 4x - 3 \)**
   - The coefficient of \(x^2\) is **negative** \( (-2) \), so the parabola opens **downward**.

3. **\( y = -(x + 1)^2 - 8 \)**
   - Expand the equation: \( y = -(x + 1)^2 - 8 = -(x^2 + 2x + 1) - 8 = -x^2 - 2x - 9 \).
   - The coefficient of \(x^2\) is **negative** \( (-1) \), so the parabola opens **downward**.

4. **\( y = 2x^2 + 8x - 3 \)**
   - The coefficient of \(x^2\) is **positive** \( (2) \), so the parabola opens **upward**.

5. **\( y = (x - 2)^2 + 1 \)**
   - Expand the equation: \( y = (x - 2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5 \).
   - The coefficient of \(x^2\) is **positive** \( (1) \), so the parabola opens **upward**.

### Summary of Openings:
1. \( y = (x + 5)^2 - 2 \): **Upward**
2. \( y = -2x^2 + 4x - 3 \): **Downward**
3. \( y = -(x + 1)^2 - 8 \): **Downward**
4. \( y = 2x^2 + 8x - 3 \): **Upward**
5. \( y = (x - 2)^2 + 1 \): **Upward**

Let me know if you'd like further details or explanations!

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Here are five follow-up questions:
1. How do you determine the vertex of a quadratic function in vertex form?
2. Can you convert a quadratic function from standard form to vertex form?
3. How does the axis of symmetry relate to the vertex in these equations?
4. What impact does the constant term have on the graph of a quadratic function?
5. Can a parabola that opens downward have a minimum point?

**Tip**: In vertex form \( y = a(x - h)^2 + k \), the sign of \(a\) determines the direction of the parabola's opening!

Determine the opening of the graph of each quadratic function.

Analyze quadratic functions to determine whether the graph opens upward or downward. Learn how the coefficient of the x^2 term affects the direction of a parabola. Suitable for students in Grades 9-11.

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