Let's break down the problem based on the provided image, which discusses various parabolas.

(a) For each parabola, determine whether it opens upward or downward:

1. $y = -\frac{1}{3}x^2$:

   • The coefficient of $x^2$ is negative ($-\frac{1}{3}$), so the parabola opens downward.

2. $y = -3x^2$:

   • The coefficient of $x^2$ is negative ($-3$), so the parabola opens downward.

3. $y = -4x^2$:

   • The coefficient of $x^2$ is negative ($-4$), so the parabola opens downward.

4. $y = \frac{1}{2}x^2$:

   • The coefficient of $x^2$ is positive ($\frac{1}{2}$), so the parabola opens upward.

(b) Choose the parabola with the narrowest graph:

• The narrowness of a parabola depends on the absolute value of the coefficient of $x^2$. The larger the absolute value, the narrower the parabola.
  • $y = -4x^2$ has the largest absolute value coefficient (4), so it is the narrowest graph.

(c) Choose the parabola with the widest graph:

• The widest parabola corresponds to the smallest absolute value of the coefficient of $x^2$.
  • $y = \frac{1}{2}x^2$ has the smallest absolute value coefficient ($\frac{1}{2}$), so it is the widest graph.

Would you like further explanations for any of these steps, or do you have any specific questions?

Here are 5 related questions you could explore:

1. How does the sign of the coefficient affect the direction of a parabola?
2. Why does the absolute value of the coefficient determine the width of the parabola?
3. How do you calculate the vertex of each parabola?
4. What is the axis of symmetry for each of these parabolas?
5. How would you graph each parabola using the coefficients provided?

Tip: A negative coefficient for $x^2$ always makes the parabola open downward, while a positive coefficient makes it open upward.