Graph of the power function y = xⁿ

The value of the power aⁿ depends on its base a as well as on the exponent n. To start with, let n be constant and a vary, then the power depends on the base alone. y = xⁿ is the power function.

In graphical representations, unless stated otherwise, assume always that both axes employ the same scale. If the scales differ, you obtain distorted curves and some of the following geometrical results lose their validity.

The line y = x¹ (angle bisector)

You have already met with the function y = x in Graphical representation of the function y = mx. Its image is a straight line. If you draw for an arbitrary point B its coordinates x and y, the triangle OAB is isosceles, because y = x, whence the base angles are 45º and the line bisects the angle XOY.

The parabola y = x².

In order to graph this curve, you make a table and enter corresponding values into the coordinate system, as shown. You must enter enough points to obtain an accurate graph!

I. Position: The curve is confined to the upper half plane.

II. Axial symmetry: Two opposite, equal values, for example, 2, correspond to the same ordinate 4. If you flip the right branch of the curve about the ordinate axis, the point A₁ moves to the point A₂: The curve is symmetric with respect to the Y-axis. The point 0 is symmetric to itself and is called the vertex.

III. Inclination: y and x do not have the same rates of increase. If |x| increases all the time by the same amount, y does not increase uniformly, but becomes larger at each step. The curve's distance from the ordinate axis increases with growing |x|, while the curve turns towards the ordinate.

The curve is called a parabola of second degree or also just a parabola. If you have a well drawn image of it on graph paper, you can extract from it the squares of numbers, for example, you find for x = 3 the square y = 9.

The cubic parabola y = x³.

I. Position: Since (+x)³ is always positive, (-x)³ always negative, you find that for x > 0, y > 0, for x < 0, y < 0. The curve passes through the first and third quadrants.

II. Symmetry: If you select two points with opposite equal abscissae, say x = 2, the corresponding ordinates are also opposite, that is, y = 8. The positive branch of the curve, on rotation about 0 by 180º, covers its negative branch, it is centri-symmetric.

III. Inclination: Its steepness is similar to that of the quadratic parabola. At the point O, it changes its curvature; such a point is called a turning point or point of inflection.

The curve is called a parabola of third degree or a cubic parabola. You can read off this curve the cubes of numbers.

Parabolas of higher order

You can also go to larger powers and draw their parabolas.

Parabolas of even order lie in the first and second quadrants, are axi-symmetric and share the points (0,0), (+1,+1) and (-1,+1), those of odd order lie in the first and third quadrants, are centri-symmetric and share the points (+1,+1), (0,0) and (-1,-1).

Exercises:

Answer

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