To create an equation that models the transformation of \( y = 2x^2 \) to the function in the graph, we need to analyze the changes that have occurred to the original function.
- Vertical Stretch: The coefficient 2 remains, indicating a vertical stretch of factor 2, which matches our original function \( y = 2x^2 \).
- Horizontal Shift: Depending on where the vertex of the parabola has moved, this will determine the horizontal shift.
- Vertical Shift: If the vertex has moved up or down, we will need to factor that into our equation.
Without seeing the specific graph, I can guide you through the reasoning:
If the parabola's vertex has moved to the right by 4 units, the equation becomes \( y = 2(x - 4)^2 \).
If it moved to the left by 4 units, it becomes \( y = 2(x + 4)^2 \).
A vertical translation up by 4 units results in \( y = 2x^2 + 4 \), while a vertical translation down by 4 units results in \( y = 2x^2 - 4 \).
To conclude which equation is correct, you must match the transformations (horizontal and vertical shifts) observed in the graph.
Based on your question, here are potential equations with their meanings:
- \( y = 2(x - 4)^2 \) indicates a right shift of 4 units.
- \( y = 2(x + 4)^2 \) indicates a left shift of 4 units.
- \( y = 2x^2 - 4 \) suggests a downward shift of 4 units.
- \( y = 2x^2 + 4 \) suggests an upward shift of 4 units.
Choose the equation that matches the transformation observed in the graph.