Reflection: a = -1.
A comparison of the curves y = xn and y = -xn shows that for equal abscissae x the ordinates assume opposite, but equal values.
The curve y = -xn is obtained from the curve y = xn by reflection in the y-axis, where n can be any integer and 0.
Since for odd n, these function yield for equal ordinates equal, but opposite abscisae, one can be obtained from the other by reflection in the y-axis.
The function y = axn increases or decreases every value of y = xndepending on whether the absolute value |a| is < or > 1. If you divide y'= x' n by y = xn, you have for the same abscissa y'/y = a/1. This proportion allows you to construct with the aid of the curve y = xn any points of the curve y = ax' n.
Let P1 be any point of the curve y = xn. The line linking P1 to A cuts the abscissa at R1. The line R1B intersects P1Q1 at the required point P'1, since by the projection rule Q1P1'/Q1P1 = OB/OA or y'/y = a/1. In the figures above, the same construction is repeated for another point.
If a < 0, OB must be drawn downwards. Reflection occurs also in this general case. If P1A is parallel to the abscissa, that is P1Q1 = 1, then P1B must also be parallel to this axis, since P1Q1 must equal a. If the value is unsuitable for the drawing, you can replace a/1 by another value, as long as the value of the quotient remains the same. Frequently, such a drawing is more easily done, if the segment OAB with its proportion a/1 is replaced by another one on a perpendicular to the abscissa with the same ratio a/1. The second figure above executes this case. You join P1 to another arbitrary point on the hyperbola P2 and obtain R2. The line RP1 intersects Q2P2 at the required point P2; by the proportion rule, y'2/y2 = y'1/y1 = a/1.
The
coefficient a changes the steepness of the curves, which is characterized by it. We talk here of stretching of curves in the
direction of the y-axis. This type of
stretching is not the same as the stretching in Geometry with
respect to a point. Here you have stretching from the x-axis
along the y-axis. The word stretching covers here all cases of shortening (|a|<1) and lengthening (|a|>1) and identity (|a| = 1).of the ordinates (|a|<1).
The curve y = -x²/4 can be generated by
reflection of the parabola y = x²/4 or also
directly by stretching of the parabola y = x².