Part 3/6:
These patterns indicate that odd powers yield curves that extend indefinitely in both directions, while even powers produce symmetrical parabolic shapes.
Observing Graphs for Higher Powers
Utilizing graphing tools, we can visualize these functions, revealing that as ( a ) increases, the curves maintain the same basic shapes with modifications in steepness at various intervals. For instance, ( x^2 ), ( x^4 ), and ( x^6 ) form similar parabolic structures, while steepness varies with the degree of the polynomial.
Case 2: Fractional Exponents
Root Functions
Next, we explore power functions where ( a ) equals ( 1/n ) (with ( n ) as a positive integer). For example: