Polynomial Graphing | Interactive Math

Interactive polynomial

Degree n Coefficient a Coefficient b Constant c

f(x) = x³ − 3x

● curve● zeros▲ local extrema◆ inflection point

Real zeros—

Local maximum—

Local minimum—

Inflection / reflection point—

Automatic x-range—

Automatic y-range—

Automatic range: the graph first finds the important points and then adds a margin so the real zeros, turning points and inflection point are visible. “Reflection point” is shown using the standard calculus term inflection point.

Explore Quadratic and Cubic Polynomial Graphs

Change the coefficients of a polynomial and observe how the shape of its graph changes. Explore quadratic and cubic functions and investigate their roots, turning points, maxima, minima and inflection points.

Use the interactive graph to connect the algebraic equation of a polynomial with its graphical behaviour.

import numpy as np
import matplotlib.pyplot as plt

# f(x) = a*x**n + b*x + c
n = 3
a = 1.0
b = -3.0
c = 0.0

# Polynomial coefficient array:
# [a, 0, 0, ..., b, c]
coef = [a] + [0]*(n-2) + [b, c]

# Real zeros
roots = np.roots(coef)
real_zeros = sorted(
    r.real for r in roots
    if abs(r.imag) < 1e-8
)

# First derivative:
# f'(x) = n*a*x**(n-1) + b
d1 = np.polyder(coef)
critical = np.roots(d1)
critical = [
    r.real for r in critical
    if abs(r.imag) < 1e-8
]

# Second derivative classifies
# local maxima and minima.
d2 = np.polyder(d1)

maxima = []
minima = []

for x0 in critical:
    y0 = np.polyval(coef, x0)
    curvature = np.polyval(d2, x0)

    if curvature < 0:
        maxima.append((x0, y0))
    elif curvature > 0:
        minima.append((x0, y0))

# Inflection points are real roots
# of f'' where concavity changes.
infl_roots = np.roots(d2)
inflections = []

for r in infl_roots:
    if abs(r.imag) < 1e-8:
        x0 = r.real
        # Check concavity on both sides
        eps = 1e-4
        left = np.polyval(d2, x0-eps)
        right = np.polyval(d2, x0+eps)

        if left * right < 0:
            inflections.append(
                (x0, np.polyval(coef, x0))
            )

print("Polynomial:", coef)
print("Real zeros:", real_zeros)
print("Maxima:", maxima)
print("Minima:", minima)
print("Inflection:", inflections)

# Choose a graph range containing
# the important x-coordinates.
important_x = (
    real_zeros
    + [x for x, y in maxima]
    + [x for x, y in minima]
    + [x for x, y in inflections]
)

if important_x:
    M = max(2, max(abs(x) for x in important_x))
else:
    M = 5

x = np.linspace(-1.4*M, 1.4*M, 1000)
y = np.polyval(coef, x)

plt.axhline(0)
plt.axvline(0)
plt.plot(x, y)
plt.grid()
plt.show()

Polynomial Graphing Python Model

This Python model calculates and plots polynomial functions. Change the coefficients and compare the resulting roots, turning points and graph shape with the interactive polynomial explorer.