Enter a function — the calculator performs the full textbook analysis: domain, parity, periodicity, zeros and sign, asymptotes, the derivative and extrema, the second derivative and inflection points, a sign table and the final graph.
Curve sketching checklist
- Domain: denominator ≠ 0, even roots of non-negative values, logarithms of positive values, excluded points of tan and cot.
- Parity: f(−x) = f(x) — even (symmetric about the y-axis); f(−x) = −f(x) — odd (symmetric about the origin).
- Periodicity: sin and cos have period 2π, tan and cot — π; sin(kx) — 2π/k.
- Zeros and sign: solve f(x) = 0 and mark where f > 0 and f < 0.
- Asymptotes: vertical x = a if a one-sided limit at a is infinite; slant y = kx + b if k = lim f(x)/x and b = lim (f(x) − kx) are finite as x → ±∞; k = 0 gives a horizontal asymptote.
- First derivative: critical points (f′ = 0 or undefined), increasing where f′ > 0, decreasing where f′ < 0; + to − is a maximum, − to + a minimum.
- Second derivative: f″ > 0 — concave up, f″ < 0 — concave down; where f″ changes sign — an inflection point.
- The table collects the signs of f′ and f″ on each interval and guides the sketch.
Exact and approximate values
Derivatives are found symbolically. Equations for polynomials and rational functions are solved via numerator roots: rational roots exactly, a quadratic remainder via the discriminant. Other equations are solved numerically; recognizable numbers (π/4, e, 1/e) are written exactly, the rest with ≈.
FAQ
How do I find a slant asymptote?
Compute k = lim f(x)/x and b = lim (f(x) − kx) as x → +∞ and separately as x → −∞. If both are finite, the asymptote is y = kx + b; k = 0 gives a horizontal one.
What is the difference between a critical point and an extremum?
A critical point is where f′ = 0 or f′ is undefined. It is an extremum only if f′ changes sign: for x³ at zero f′ = 0 but the sign does not change.
What are concavity and inflection points?
If f″ > 0 the graph is concave up (above its tangents), if f″ < 0 — concave down. An inflection point is where f″ changes sign and f is defined.
Why are some values marked ≈?
Equations like x·eˣ = 1 or cos x = x have no closed-form solution — roots are found numerically to about 10⁻¹². Simple numbers (fractions, √n, multiples of π, powers of e) are recognized and written exactly.
How is a periodic function analyzed?
Find the period (sin, cos — 2π, tan — π) and analyze one period; zeros, extrema and inflection points repeat with the period. A periodic function has no asymptotes at infinity.