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When transforming f(x)=x3 to f(x)=−(23x+3)3+2 , what transformations should be performed and in what order?(1 point)

Responses

horizontal stretch, translated horizontally to the left, and reflected across the x-axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the x -axis, then translated vertically up

horizontal stretch, translated horizontally to the left, and reflected across the y-axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the y -axis, then translated vertically up

horizontal stretch, translated horizontally to the right, and reflected across the x-axis, then translated vertically up
horizontal stretch, translated horizontally to the right, and reflected across the x -axis, then translated vertically up

translated horizontally to the left, horizontal stretch, and reflected across the x-axis, then translated vertically up
translated horizontally to the left, horizontal stretch, and reflected across the x -axis, then translated vertically up
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1 answer

To transform \( f(x) = x^3 \) into \( f(x) = -(23x + 3)^3 + 2 \), we need to break it down step by step:

  1. Horizontal Stretch and Translation: The expression \( 23x + 3 \) can be rewritten as \( 23(x + \frac{3}{23}) \). The coefficient 23 indicates a horizontal compression (since 23 is greater than 1). The shift of \( +3 \) indicates a translation to the left by \( \frac{3}{23} \).

  2. Reflection Across the X-axis: The leading negative sign indicates a reflection across the x-axis.

  3. Vertical Translation: The \( +2 \) at the end indicates a vertical shift upwards by 2 units.

In summary, the correct order of transformations would be:

  • Reflect across the x-axis
  • Translate horizontally to the left
  • Apply a horizontal compression (not stretch in this case) by a factor of \( \frac{1}{23} \)
  • Translate vertically upwards

Thus, the closest response option would be: translated horizontally to the left, horizontal stretch, and reflected across the x-axis, then translated vertically up.