Final answer:
The function y = −3(x + 5)² − 7 is a transformed parabola with a vertical stretch by a factor of 3, reflection across the x-axis, horizontal translation 5 units left, and downward translation of 7 units.
Step-by-step explanation:
The transformation of the function y = −3(x + 5)² − 7 involves several steps. The function can be analyzed by identifying each part of the equation and its effect on the graph of the function. The transformation is applied to the parent function y = x², which is a parabola opening upwards.
The factor −3 before the squared term indicates a vertical stretch by a factor of 3 and a reflection across the x-axis, making the parabola open downwards. The (x + 5) inside the squared term represents a horizontal translation of 5 units to the left. Lastly, the − 7 after the squared term translates the parabola downwards by 7 units along the y-axis.
Combining these transformations, the final graph of the function is a downward opening parabola that has been stretched vertically, moved left by 5 units and down by 7 units from the origin. This transformed function is different from even and odd functions as it does not exhibit symmetry about the y-axis nor does it have rotational symmetry about the origin.