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The parabola y=x^(2) is shifted down by 3 units and to the left by 2 units. What is the equation of the new parabola? y=

The parabola y=x2 y=x^{2} is shifted down by 33 units and to the left by 22 units.\newlineWhat is the equation of the new parabola?\newliney= y=

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Q. The parabola y=x2 y=x^{2} is shifted down by 33 units and to the left by 22 units.\newlineWhat is the equation of the new parabola?\newliney= y=
  1. Shift down by 33 units: To shift the parabola y=x2y = x^2 down by 33 units, we subtract 33 from the yy-value. This gives us the equation y=x2−3y = x^2 - 3.
  2. Shift left by 22 units: To shift the parabola to the left by 22 units, we replace xx with (x+2)(x + 2) in the equation. This gives us the equation y=(x+2)2−3y = (x + 2)^2 - 3.
  3. Expand and simplify: Now we expand the equation y=(x+2)2−3y = (x + 2)^2 - 3 to check for any simplification. Expanding (x+2)2(x + 2)^2 gives x2+4x+4x^2 + 4x + 4. So the equation becomes y=x2+4x+4−3y = x^2 + 4x + 4 - 3.
  4. Final equation: Simplify the equation by combining like terms. The constant terms 44 and −3-3 combine to give 11. So the final equation of the new parabola is y=x2+4x+1y = x^2 + 4x + 1.

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