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© Copyright 2003, Jim Loy
Here are a few algebraic curves. Each of these can be of different sizes, and some can be of different shapes, depending upon the specific parameters of the equations.
Above are a semicircle (equation y = sqrt(1-x^2)), half of an ellipse (y = sqrt(1-x^2/9)), and a right hyperbola (y = 1/x). These are functions; each x value produces only one y value. In rectangular coordinates, a circle would be the graph of two functions, y = sqrt(1-x^2) and y = -sqrt(1-x^2).
Above are a parabola (y = x^2), a cubic parabola (y = x^3), and a fourth degree equation (y = x^4-2x^3-x^2-x+2).
Here are the sine (y = sin(x)) and cosine (y = cos(x)) functions. They are of the same shape, but of different phase.
These are the tangent function (y = tan(x)) and the cosecant function (y = csc(x) = 1/sin(x)).
Here is y = e^x and y = ln(x). Ln is the abbreviation for the natural logarithm, and it is the inverse function of y = e^x.
And these are the absolute value (y = |x|) and the normal probability curve.