#!/usr/bin/env python3 """Graphs of equations: the graph is a set of points, intercepts and symmetry are tests on it. Run: python3 graphs_intercepts_symmetry.py The graph of an equation in x and y is the set of all points (x, y) whose coordinates make the equation true. Everything on this page follows from that one sentence: on the graph? substitute the point and see if the equation holds x-intercepts the graph's points with y = 0, so set y = 0 and solve y-intercepts the graph's points with x = 0, so set x = 0 and solve symmetry a mirror image of every point is also on the graph Each equation is written as F(x, y) = 0. Numbers are fractions, so "the equation holds" means exactly. The pictures show only the points with whole number coordinates that satisfy the equation exactly as *; between them the graph is a smooth curve, and the grid marks each square it passes through. """ from fractions import Fraction as F EQUATIONS = [ # name, F(x, y) with the graph as F = 0 ("y = x^2 - 4", lambda x, y: x * x - 4 - y), ("x = y^2", lambda x, y: y * y - x), ("y = x^3 - x", lambda x, y: x ** 3 - x - y), ("x^2 + 4y^2 = 16", lambda x, y: x * x + 4 * y * y - 16), ("y = x + 2", lambda x, y: x + 2 - y), ("y = x^2 + 1", lambda x, y: x * x + 1 - y), ] R = 6 # the grid runs from -R to R SAMPLES = [F(n, d) for n in range(-7, 8) for d in (1, 2, 3)] def crosses(f, x: int, y: int) -> bool: """Does the curve F = 0 pass through the unit cell centred on (x, y)? It does if F is 0 somewhere on the cell's corners or centre, or takes both signs among them: F is a polynomial, so it cannot change sign without passing through 0 in between. """ h = F(1, 2) vals = [f(F(x) + dx, F(y) + dy) for dx in (-h, 0, h) for dy in (-h, 0, h)] return 0 in vals or (min(vals) < 0 < max(vals)) def plot(f) -> list[str]: rows = [] for y in range(R, -R - 1, -1): cells = [] for x in range(-R, R + 1): if f(F(x), F(y)) == 0: cells.append("*") elif crosses(f, x, y): cells.append("o") elif x == 0 and y == 0: cells.append("+") elif x == 0: cells.append("|") elif y == 0: cells.append("-") else: cells.append(".") rows.append(f" {y:>3} " + "".join(f"{c:>3}" for c in cells)) rows.append(" " + "".join(f"{x:>3}" for x in range(-R, R + 1)) + " x") return rows def roots(g) -> list[F]: """Solutions of g(t) = 0 among fractions with small numerator and denominator.""" found = {F(n, d) for n in range(-20, 21) for d in (1, 2, 3, 4) if g(F(n, d)) == 0} return sorted(found) def symmetric(f, mirror) -> bool: """Replacing (x, y) by its mirror image gives an equivalent equation: F' = F or F' = -F.""" same = all(f(*mirror(x, y)) == f(x, y) for x in SAMPLES for y in SAMPLES) flip = all(f(*mirror(x, y)) == -f(x, y) for x in SAMPLES for y in SAMPLES) return same or flip MIRRORS = [ ("x-axis", "replace y by -y", lambda x, y: (x, -y)), ("y-axis", "replace x by -x", lambda x, y: (-x, y)), ("origin", "replace both", lambda x, y: (-x, -y)), ] def fmt(x: F) -> str: return str(x.numerator) if x.denominator == 1 else f"{x.numerator}/{x.denominator}" def main() -> None: name, f = EQUATIONS[0] print(f"1. A GRAPH IS A SET OF POINTS: {name}") for line in plot(f): print(line) print(" * is a whole-number point (x, y) that makes the equation true exactly;") print(" o is a square the curve passes through between such points.") print() print("2. IS THE POINT ON THE GRAPH? SUBSTITUTE AND SEE") for p in [(2, 0), (1, -3), (1, -2), (-3, 5), (0, 4)]: x, y = F(p[0]), F(p[1]) lhs, rhs = y, x * x - 4 verdict = "on the graph" if lhs == rhs else "not on it" print(f" ({p[0]:>2}, {p[1]:>2}): y = {fmt(lhs):>2}, x^2 - 4 = {fmt(rhs):>2} {verdict}") print(" (0, 4) fails although (0, -4) passes: a sign is a different point.") print() print("3. INTERCEPTS: SET THE OTHER COORDINATE TO 0, THEN SOLVE") print(f" {'equation':18}{'x-intercepts (y = 0)':>24}{'y-intercepts (x = 0)':>24}") for name, f in EQUATIONS: xs = roots(lambda t: f(t, F(0))) ys = roots(lambda t: f(F(0), t)) show = lambda v: ", ".join(fmt(t) for t in v) if v else "none" print(f" {name:18}{show(xs):>24}{show(ys):>24}") print(" An intercept is a single number, a coordinate: the x-intercepts of") print(" y = x^2 - 4 are -2 and 2, and the points are (-2, 0) and (2, 0).") print(" y = x^2 + 1 never reaches the x-axis: x^2 = -1 has no real solution.") print(" x = y^2 has 0 as both: the graph goes through the origin.") print() print("4. SYMMETRY: DOES THE MIRROR IMAGE OF THE EQUATION SAY THE SAME THING?") print(f" {'equation':18}" + "".join(f"{m[0]:>10}" for m in MIRRORS)) for name, f in EQUATIONS: print(f" {name:18}" + "".join(f"{('yes' if symmetric(f, m[2]) else 'no'):>10}" for m in MIRRORS)) print(" x-axis: replace y by -y. y-axis: replace x by -x. origin: replace both.") print(" If the new equation is equivalent to the old one, the graph has that symmetry.") print() print("5. THE TEST, BY HAND, ON y = x^3 - x") print(" origin: -y = (-x)^3 - (-x) = -x^3 + x; multiply by -1: y = x^3 - x. Same: symmetric.") print(" y-axis: y = (-x)^3 - (-x) = -x^3 + x. Different equation: not symmetric.") f = EQUATIONS[2][1] print(" A witness for 'not': (2, 6) is on the graph, its y-axis mirror (-2, 6) is not:", f"F(2, 6) = {fmt(f(F(2), F(6)))}, F(-2, 6) = {fmt(f(F(-2), F(6)))}.") print() print("6. ANY TWO SYMMETRIES FORCE THE THIRD") print(" Reflecting in the x-axis and then the y-axis is a half turn about the origin:") print(" (x, y) -> (x, -y) -> (-x, -y). So x^2 + 4y^2 = 16, with both axis symmetries,") print(" gets origin symmetry for free. The ellipse:") for line in plot(EQUATIONS[3][1]): print(line) print() print("7. A GRAPH OF y = f(x) WITH x-AXIS SYMMETRY IS FLAT") print(" If (x, y) and (x, -y) are both on it, one x has two y's unless y = 0.") print(" So among graphs of functions, only y = 0 is symmetric about the x-axis;") print(" x = y^2 is symmetric about it precisely because y is not a function of x there.") if __name__ == "__main__": main()