Based on the uploaded image, the problem involves analyzing the transformations applied to the graph of $y = x^2$ to match the black graphs shown in each subfigure (a, b, c, d). I'll describe the transformations and equations for each case:

Analysis

a)

• The green graph $y = x^2$ is transformed into the black graph.
• Transformation: The graph is reflected over the x-axis and shifted up by 12 units.
• Equation: $y = -x^2 + 12$.

b)

• The green graph $y = x^2$ is transformed into the black graph.
• Transformation: The graph is reflected over the x-axis and shifted down by 8 units.
• Equation: $y = -x^2 - 8$.

c)

• The green graph $y = x^2$ is transformed into the black graph.
• Transformation: The graph is shifted 2 units to the right and up by 2 units.
• Equation: $y = (x - 2)^2 + 2$.

d)

• The green graph $y = x^2$ is transformed into the black graph.
• Transformation: The graph is reflected over the x-axis, shifted 2 units to the right, and shifted down by 4 units.
• Equation: $y = -(x - 2)^2 - 4$.

Let me know if you'd like more details or explanations on transformations!

Expanding Understanding

1. How can you describe vertical shifts for general functions like $f(x)$?
2. What is the general form of a parabola and how do horizontal shifts affect it?
3. What happens when a graph is reflected over the x-axis?
4. How does combining multiple transformations (e.g., shifts and reflections) affect the graph?
5. Can transformations ever change the "shape" of a parabola?

Tip

When analyzing transformations, follow this sequence: reflections, horizontal shifts, vertical shifts. This makes it easier to organize your approach!