Graphs of functions

What 31 common functions look like, and why. Each page has the graph, its domain and range, intercepts and turning points computed to full precision, a table of values, transformations and a method for sketching it by hand.

Powers, roots and polynomials

Graph of a linear function: y = mx + b

A straight line: slope m, y-intercept b.

Graph of x² (the parabola y = x²)

The basic parabola: vertex (0, 0), opens upward.

Graph of x³ (the cubic y = x³)

An S-shape through the origin, flat at its inflection point.

Square root graph: y = √x

Starts at the origin; half of a sideways parabola.

Graph of the cube root: y = ∛x

Defined for all x; vertical tangent at the origin.

Rational functions and absolute value

Absolute value graph: y = |x|

A V with its vertex at the origin, slopes ±1.

Graph of 1/x (the reciprocal function)

A hyperbola with asymptotes along both axes.

Graph of 1/x²

Always positive; both branches above the x-axis.

Trigonometric functions

Sine graph: y = sin x

Period 2π, amplitude 1, zeros at every multiple of π.

Graph of cosine: y = cos x

The sine wave shifted left by π/2; starts at its peak.

Tan graph: y = tan x

Period π, asymptotes at π/2 + nπ, always increasing.

Graph of cotangent: y = cot x

Period π, asymptotes at multiples of π, always decreasing.

Graph of secant: y = sec x

1/cos x: U-shaped branches, never between −1 and 1.

Graph of cosecant: y = csc x

1/sin x: asymptotes at every multiple of π.

Inverse trigonometric functions

Graph of arcsin x (inverse sine)

Domain [−1, 1], range [−π/2, π/2]; sin x reflected in y = x.

Graph of arccos x (inverse cosine)

Domain [−1, 1], range [0, π], always decreasing.

Arctan graph: y = arctan x (inverse tangent)

All real inputs, outputs between −π/2 and π/2.

Exponential and logarithmic functions

Graph of ln x (natural logarithm)

Defined for x > 0, through (1, 0) and (e, 1).

Graph of log x (base 10 logarithm)

Base 10: through (1, 0) and (10, 1); every log graph has this shape.

Graph of e^x

Through (0, 1), always positive; its slope equals its height.

Graph of an exponential function: y = aˣ

y = aˣ: growth when a > 1, decay when 0 < a < 1.

Graph of exponential decay: y = (1/2)ˣ

Halves every step: (0, 1), (1, ½), (2, ¼), …

Hyperbolic functions

Graph of sinh x (hyperbolic sine)

(eˣ − e⁻ˣ)/2: odd, increasing, through the origin.

Graph of cosh x (hyperbolic cosine, the catenary)

The catenary: the shape of a hanging chain.

Graph of tanh x (hyperbolic tangent)

S-curve between −1 and 1; a classic activation function.

Other notable functions

Graph of x sin x

Oscillations that grow, trapped between y = ±x.

Graph of x cos x

An odd, growing oscillation between y = ±x.

Graph of sin x / x (the sinc function)

A decaying wave with limit 1 at the origin.

Graph of e^(−x²): the Gaussian bell curve

The bell curve: peak (0, 1), area √π.

Graph of the logistic function (sigmoid)

The S-curve from 0 to 1, midpoint (0, ½).

Graph of the floor function (greatest integer function)

A staircase: jumps of 1 at every integer.

Graph any function

Type any formula into the calculator and it is graphed exactly, with its roots, extrema and intersections marked.

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