https://file.aiquickdraw.com/mathbot/file/1730731309552ssvj1qw1.jpg

In the image, we are asked to determine the direction in which each parabola opens based on the equations provided. Let’s analyze each equation individually to confirm the answers:

1. **Equation:** \( y = 2x^2 + 6x - 1 \)
   - **Analysis:** The coefficient of \( x^2 \) is \( +2 \), which is positive. When the coefficient of \( x^2 \) in a quadratic equation is positive, the parabola opens **upward**.
   - **Answer:** Upward.

2. **Equation:** \( y = -3x^2 - 5x + 2 \)
   - **Analysis:** The coefficient of \( x^2 \) is \( -3 \), which is negative. When the coefficient of \( x^2 \) is negative, the parabola opens **downward**.
   - **Answer:** Downward.

3. **Equation:** \( y = (x - 3)^2 - 5 \)
   - **Analysis:** Expanding \( (x - 3)^2 \) gives \( x^2 \) with a positive coefficient. Since the squared term \( x^2 \) is positive, this parabola opens **upward**.
   - **Answer:** Upward.

The solutions provided in the image are correct. 

Would you like further details on any of these concepts or a breakdown of how to determine the direction for different forms of quadratic equations?

Here are some related questions that could deepen your understanding:

1. How do we determine the vertex of a parabola given a standard form equation?
2. What effect does the coefficient of \( x \) in the term \( bx \) have on the shape or position of a parabola?
3. How do we convert a quadratic equation from standard form to vertex form?
4. Why does a positive leading coefficient cause a parabola to open upward?
5. How does the value of the leading coefficient affect the width of the parabola?

**Tip:** Remember that the sign of the \( x^2 \) term (positive or negative) determines whether a parabola opens upward or downward.

Describe the opening direction of the parabolas given by the equations.

Learn how to determine if a parabola opens upward or downward by analyzing the leading coefficient in quadratic equations. Suitable for students in Grades 8-10, this example covers multiple forms of quadratic equations.

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