Final answer:
Graphing various powers of x for -1 to 1 shows increasing flatness near the y-axis as the exponent rises, with rapid increase at the extremes for even powers and asymmetry for odd powers.
Step-by-step explanation:
When graphing the functions y = x², y = x³, y = x´, and y = x⁵ for -1≤ x ≤1, we see different behavior as the exponent increases. A graph can be plotted using specific values for (x,y) data pairs.
For y = x², which is a parabola, the graph will be symmetrical across the y-axis, while for y = x³, which is a cubic function, the graph will be symmetrical with respect to the origin (0,0). The functions y = x´ and y = x⁵ will look similar to y = x² and y = x³ respectively, but will be flatter towards the y-axis and steeper away from it due to the higher powers.
The function y = x¹⁰⁰ would look even flatter near the y-axis except at the points -1 and 1 where it will rise sharply. Conversely, y = x¹⁰⁰¹ will introduce a slight tilt due to the odd power, implying asymmetric behavior around the origin. Creating a table of values confirms that, as the exponent increases, the values of y become very small until x is very close to -1 or 1.