Mth 163
What are the basic points for y=f(x)?
The answer depends on the function f(x).
The basic linear function is y = x and its basic points are the x = 0 and x = 1 points.
The basic quadratic function is y = x^2 and its basic points are the x = -1, x = 0 and x = 1 points.
The basic points of the exponential function y = 2^x are the x = 0 and x = 1 points, and also the horizontal asymptote at the negative x axis. The asymptote isn't really a point but we include it among the three basic things we need in order to quickly graph an exponential function.
The basic points of a power function y = x^p are the x = -1, x = 0, x = 1/2, x = 1 and x = 2 points. If p is negative, then the x = 0 'point' isn't really a point but a vertical asymptote, and as the trend of basic points shows, for negative p the x axis is a horizontal asymptote in both directions.
If a function is of the form y = A f(x-h) + k then the basic points (and all points) are 'stretched' to positions A times as far from the x axis, horizontally shifted h units and vertically shifted k units. By applying this principle just to the basic points, we can quickly and easily locate the graph of A f(x-h) + k for any of our basic functions.
The basic points of y = A f(x-h) + k are the points you get when you apply these stretches and shifts to the basic points of the function f(x).