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With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function.
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing. When f {\displaystyle f} is a strictly monotonic function, then f {\displaystyle f} is injective on its domain, and if T {\displaystyle T} is the range of f {\displaystyle f} , then there is an inverse function on T ...
Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree. The first degree polynomial equation = + is a line with slope a. A line will connect any two ...
Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.
Power functions – relationships of the form = – appear as straight lines in a log–log graph, with the exponent corresponding to the slope, and the coefficient corresponding to the intercept. Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most ...
If we draw a graph of the logistic map (), we can observe that the slope of the tangent at the fixed point exceeds 1 at the boundary = and becomes unstable. At the same time, two new intersections appear, which are the periodic points x f 1 ( 2 ) {\displaystyle x_{f1}^{(2)}} and x f 2 ( 2 ) {\displaystyle x_{f2}^{(2)}} .