The image contains quadratic function problems involving graphing and word problems. Let’s break down the tasks:

1. Graph each function (Problems 1 and 2):

   • For $y = 4x^2$: A parabola opening upwards, passing through points like (0,0), (1,4), (-1,4), etc.
   • For $y = -3x^2$: A parabola opening downwards, passing through points like (0,0), (1,-3), (-1,-3), etc.

2. Problem 3:

   • A penny dropped from a height follows $d = -16t^2 + 196$.
   • To determine the time it takes to hit the water, set $d = 0$: $-16t^2 + 196 = 0$ Solving this gives $t = 3.5$ seconds.

3. Problem 4:

   • The equation $A = 0.25d^2$ represents the area of the movie screen in terms of distance $d$ from the projector.
   • To estimate $d$ when $A = 7$, solve $0.25d^2 = 7$: $d^2 = 28 \implies d = \sqrt{28} \approx 5.29$

4. Problem 5a:

   • To represent the area $A$ of the rectangle in terms of $x$: $A = x(12 - x) = 12x - x^2$

5. Problem 5b:

   • The maximum area occurs at the vertex of the parabola $A = 12x - x^2$.
   • The x-coordinate of the vertex is: $x = -\frac{b}{2a} = -\frac{12}{2(-1)} = 6$ So, the dimensions are $6 \times 6$, and the maximum area is $36 \, \text{square inches}$.

Let me know if you'd like detailed steps for any of these problems!

Follow-Up Questions:

1. How do we derive the time for Problem 3 step-by-step?
2. Can we explore how the graph of $y = -3x^2$ relates to the vertex form?
3. What are the units of the roots in Problem 3, and why is only one value used?
4. How do we derive $d^2 = 28$ in Problem 4?
5. Can the maximum area in Problem 5 be shown graphically?

Tip: Always verify the vertex formula $x = -b/2a$ for maximizing or minimizing quadratic functions.