Use the image to answer the question.

Question

A parabola is plotted in a coordinate plane. The x-axis ranges from negative 4 to 4 and the y-axis ranges from negative 6 to 6 in one-unit increments.

Create an equation that models the transformation of y=2x2
to the function in the graph.

(1 point)
Responses

y=2(x−4)2
y equals 2 left parenthesis x minus 4 right parenthesis sqaured

y=2(x+4)2
y equals 2 left parenthesis x plus 4 right parenthesis sqaured

y=2x2−4
y equals 2 x squared minus 4

y=2x2+4
y equals 2 x squared plus 4
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Answer 1

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.

1. Vertical Stretch: The coefficient 2 remains, indicating a vertical stretch of factor 2, which matches our original function $y=2x^2$.
2. Horizontal Shift: Depending on where the vertex of the parabola has moved, this will determine the horizontal shift.
3. 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.