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.