Which Steps Transform The Graph Of Y = X^2 To Y = 2x^2?

Which Steps Transform The Graph Of Y X2 To Y 2x 22 2 1 P

Which steps transform the graph of y = x2 to y = –2(x – 2)2 + 2? (1 point) translate 2 units to the left, translate down 2 units, and stretch by the factor 2 translate 2 units to the right, translate up 2 units, and stretch by the factor 2 reflect across the x-axis, translate 2 units to the left, translate down 2 units, and stretch by the factor 2 reflect across the x-axis, translate 2 units to the right, translate up 2 units, and stretch by the factor 2

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The transformation of the quadratic function y = x2 into the form y = –2(x – 2)2 + 2 involves several geometric transformations on its graph. To understand these transformations, it's instructive to dissect the changes step by step, relating each to familiar geometric operations such as translations, reflections, and stretches or compressions.

Starting with the basic parabola y = x2, which opens upward with its vertex at the origin (0,0), the goal is to comprehend how the modifications to produce y = –2(x – 2)2 + 2 alter its graph. The transformations can be broken down into a sequence involving horizontal shifts, vertical shifts, reflections, and vertical stretches or compressions.

Horizontal Shift

In the original transformation, the argument of the squared term changes from x to (x – 2). Since the general form y = (x – h)2 translates the graph h units horizontally, when h is positive, the shift is to the right. Here, we have (x – 2), indicating a shift 2 units to the right. This moves the vertex from (0, 0) to (2, 0).

Vertical Reflection and Stretch

Next, the coefficient of the squared term is –2. The negative sign reflects the graph across the x-axis, turning the parabola downward. The factor 2 indicates a vertical stretch by a factor of 2; that is, the parabola becomes narrower, with its arms steeper than the original.

Vertical Shift

Finally, the +2 outside the squared term shifts the graph up by 2 units. This relocation moves the vertex from (2, 0) to (2, 2), positioning the transformed parabola with its vertex at (2, 2).

Summary of Sequential Transformations

  1. Translate the graph 2 units to the right, corresponding to the (x – 2) inside the squared term.
  2. Reflect the graph across the x-axis due to the negative coefficient, turning the upward-opening parabola downward.
  3. Stretch vertically by a factor of 2, making the parabola steeper and narrower.
  4. Translate the graph 2 units upward to account for the +2 outside the squared term.

Matching with Multiple Choice Options

The correct sequence aligns with the option: reflect across the x-axis, translate 2 units to the right, translate up 2 units, and stretch by the factor 2. This sequence correctly encapsulates all the transformations involved: the reflection, horizontal shift, vertical shift, and the vertical stretch.

Conclusion

Understanding the transformations from y = x2 to y = –2(x – 2)2 requires breaking down the modifications into their geometric counterparts. This approach clarifies how each step changes the parabola's position and shape, and ensures accurate graphing and interpretation of quadratic functions.

References

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