Inflection Points of Graph

inflection_points(equation_type, coefficients, precision=4)

Calculates the inflection points of a specific function

Parameters

• equation_type (str) – Name of the type of function for which inflections must be determined (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients to use to generate the equation to investigate

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the results

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• ValueError – Last argument must be a positive integer

Returns

points – Values of the x-coordinates at which the original function has an inflection point; if the function is sinusoidal, then only five results within a two-period interval will be listed, but a general form will also be included; if the function has no inflection points, then it will return a list of None

Return type

list of float or str

See also

• Roots for key functions: linear_roots(), quadratic_roots(), cubic_roots(), hyperbolic_roots(), exponential_roots(), logarithmic_roots(), logistic_roots(), sinusoidal_roots()

• Graphical analysis: critical_points(), sign_chart(), key_coordinates()

Notes

• Critical points for the second derivative of a function: \(c_i = \{ c_1, c_2, c_3, \cdots, c_{n-1}, c_n \}\)

• X-coordinates of the inflections of the function: \(x_{infl} = \{ x \mid x \in c_i, \left( f''(\frac{c_{j-1} + c_j}{2}) < 0 \cap f''(\frac{c_j + c_{j+1}}{2}) > 0 \right) \\ \cup \left( f''(\frac{c_{j-1} + c_j}{2}) > 0 \cap f''(\frac{c_j + c_{j+1}}{2}) < 0 \right) \}\)

• Inflection Points

Examples

Import inflection_points function from regressions library

>>> from regressions.analyses.inflections import inflection_points

Calculate the inflection points of a cubic functions with coefficients 1, -15, 63, and -7

>>> points_cubic = inflection_points('cubic', [1, -15, 63, -7])
>>> print(points_cubic)
[5.0]

Calculate the inflection points of a sinusoidal functions with coefficients 2, 3, 5, and 7

>>> points_sinusoidal = inflection_points('sinusoidal', [2, 3, 5, 7])
>>> print(points_sinusoidal)
[5.0, 6.0472, 7.0944, 8.1416, 9.1888, '5.0 + 1.0472k']