Recognize a function by its graph, sketch it by hand following a fixed method, and read off the graph what happens to a function near a tricky point or as x goes to infinity. Aimed at High school and University, it assumes the student already knows the derivative (the derivada lab) and uses it here as just another tool, not as the object of study.

What is learned

The path goes from perception (recognizing a family at a glance) to procedure (sketching following a method) and ends in concepts (limits and continuity, always read off the graph with zoom, never computed algebraically).

• Rooms 1–4: review of the families — polynomials (quadratic → cubic), rationals (the hyperbola and its shifts), exponential and logarithm, and trigonometric functions (sine, cosine, tangent, arcsine, arctangent) —, with their domain, range, growth, and a recognition challenge per family.
• Room 5: a bridge room that puts all four families together and asks to classify them before the test.
• Rooms 6–7: a cross-family recognition test — from graph to formula (6 options) and from formula to graph (4 options).
• Rooms 8–13: the sketching method (domain, axis intercepts, sign, growth and extrema using f′, extra points, behavior at infinity) explained once and applied to each family, closing with an unscaffolded integrative challenge.
• Rooms 14–20: graphical limits via zoom — the limit at a point, jump discontinuity, infinite discontinuity (vertical asymptote), removable discontinuity (hole), limits at infinity, a discontinuity-classification challenge, and a final review.

Key mathematical ideas

• Every function family has a characteristic "silhouette": the shape of the curve, its domain and its range are the first clue to recognizing it without computing anything.
• Sketching a function by hand always follows the same procedure: domain → axis intercepts → sign of f → growth/decay and extrema (sign of f′) → a few extra points → behavior at infinity.
• The sign of f′ shows where a function grows (f′>0) and where it shrinks (f′<0); extrema sit where f′ is zero or doesn't exist.
• The limit of a function at a point is the value it approaches from both sides; if the one-sided limits disagree, there's a jump discontinuity.
• An infinite discontinuity happens when the function shoots off to ±∞ near a point (a vertical asymptote); a removable discontinuity happens when the limit exists but the function is undefined (or takes another value) right at that point — a "hole" in the graph.
• Limits at infinity describe what a function does as x grows or shrinks without bound: it may settle onto a horizontal asymptote, grow without bound, or approach a slanted line.