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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 plus 4 right parenthesis sqaured

y=2x2−4
y equals 2 x squared minus 4

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

y=2x2+4

1 answer

To determine the correct transformed equation of the parabola \( y = 2x^2 \), we need to consider the transformation characteristics.

  1. Horizontal Translation: This involves moving the vertex of the parabola left or right by modifying the \( x \) term in the equation.

    • The expression \( (x + 4) \) indicates a leftward shift by 4 units.
    • The expression \( (x - 4) \) indicates a rightward shift by 4 units.

  2. Vertical Translation: This involves moving the graph upwards or downwards by adding or subtracting from the entire function.

    • If we see a "+4" or "-4" after the main function, it indicates vertical shifts.

Given these transformations and the options presented:

  • Option 1: \( y = 2(x + 4)^2 \) (Left shift)
  • Option 2: \( y = 2x^2 - 4 \) (Downward shift)
  • Option 3: \( y = 2(x - 4)^2 \) (Right shift)
  • Option 4: \( y = 2x^2 + 4 \) (Upward shift)

Since the vertex of the original parabola \( y = 2x^2 \) is at (0,0), the transformed parabola needs to move, depending on the context provided by the graph. To choose the correct response based on provided transformations without more information about the exact graph, you could analyze it as below:

If the graph shows the vertex moved to the left, choose the first option. If it's shifted down, pick the second option.

Considering the transformations, without the exact position of the graph clearly stated, Option 2 \( y = 2x^2 - 4 \), commonly indicates a downward shift, which might fit the transformation if the vertex has indeed shifted downwards, but the exact transformation cannot be concluded as the graph is not visible.

Final Decision Based on Potential Downward Movement: If you choose based solely on the best generic understanding of transformations, go with Option 2. However, ensure to visually confirm the graph positions if possible.