∵ 1^a= 1

∴ Power function image must pass through the fixed point (1, 1).

A>0 a = 0 When 0, the image is at a fixed point (0,0).

When a is odd, y is odd function, which is symmetrical about the origin; When a is an even number, y is an even function and is symmetric about y.

∵Y'=aX^(a- 1)

∴ When A is a positive odd number, Y is a increasing function, and when A is a negative odd number, Y is a decreasing function (piecewise,-∞→ 0,0 → +∞).

When a is a positive even number, the negative semi-axis Y of X is a decreasing function, and the positive semi-axis Y of X is a increasing function. The negative semi-axis y of x is increasing function, and the positive semi-axis y of x is a decreasing function.

Extended data:

Power function property

1, positive attribute

When α >; 0, the power function y=xα has the following properties:

A, the images all pass through the point (1,1) (0,0);

B, the function in the image is the increasing function in the interval [0, +∞);

C, in the first quadrant, α >; 1, the derivative value increases gradually; When α= 1, the derivative is constant; 0 & ltα& lt； 1, the derivative value gradually decreases and approaches 0 (the function value increases);

2. Negative nature

When α

A, the images all pass through the point (1,1);

B, the image is a decreasing function in the interval (0, +∞); (Content supplement: If it is X-2, it is easy to get that it is an even function. Using symmetry, the symmetry axis is the Y axis, and the image can be monotonically increased in the interval (-∞, 0). The same is true for other even functions).

C, in the first quadrant, there are two asymptotes (coordinate axes), the independent variable approaches 0, the function value approaches +∞, the independent variable approaches +∞, and the function value approaches 0.