To solve this problem, we need to determine the equations for the parabolas based on the given sets of points and the transformations applied to the red parabola.

Step 1: Determine the equation of the red parabola.

The red parabola passes through the points (-1, 1), (0, 0), and (1, 1). These points suggest that the red parabola is centered at the origin, and it seems to follow a standard form of a quadratic function: $y = x^2$ This is the equation of the red parabola, which passes through the points (-1, 1), (0, 0), and (1, 1).

Step 2: Apply transformations to the red parabola to obtain the other parabolas.

Each of the other sets of points is formed by shifting the red points either vertically or horizontally. For a function $y = f(x)$, the transformations are as follows:

• Vertical shift: $y = f(x) + k$ (shifts the graph up by $k$ units if $k > 0$ or down by $k$ units if $k < 0$).
• Horizontal shift: $y = f(x - h)$ (shifts the graph right by $h$ units if $h > 0$ or left by $h$ units if $h < 0$).

Step 3: Find the equations for each parabola.

1. Blue Parabola (Shifted 3 units up)

This set of points is shifted 3 units up from the red points, so the equation for the blue parabola is: $y = x^2 + 3$

2. Green Parabola (Shifted 4 units to the right)

This set of points is shifted 4 units to the right from the red points, so the equation for the green parabola is: $y = (x - 4)^2$

3. Orange Parabola (Shifted 3 units down)

This set of points is shifted 3 units down from the red points, so the equation for the orange parabola is: $y = x^2 - 3$

4. Purple Parabola (Shifted 4 units to the left)

This set of points is shifted 4 units to the left from the red points, so the equation for the purple parabola is: $y = (x + 4)^2$

Final Equations:

• Red parabola: $y = x^2$
• Blue parabola: $y = x^2 + 3$
• Green parabola: $y = (x - 4)^2$
• Orange parabola: $y = x^2 - 3$
• Purple parabola: $y = (x + 4)^2$

These equations describe the five parabolas, each passing through the respective points after the specified transformations.

Would you like more details or have any questions about this solution? Here are 5 related questions for further exploration:

1. How does the equation of a parabola change when it is reflected across the x-axis?
2. What is the effect of shifting a quadratic function horizontally by $h$ units?
3. Can you find the vertex of the purple parabola $y = (x + 4)^2$?
4. How would the equation change if we shifted the red parabola both vertically and horizontally?
5. What is the general form of a quadratic function, and how do transformations affect it?

Tip: When working with transformations, remember that horizontal shifts are applied inside the function, while vertical shifts are applied outside the function.