Teaching objectives
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). Limits are first read off the graph, by moving a point in from each side or zooming out, and then worked out by hand in short algebraic challenges: substituting when the function is continuous, factoring and simplifying a 0/0, comparing degrees at infinity.
- Rooms 1–5: review of the families — polynomials (quadratic → cubic), rationals (the hyperbola and its shifts), exponential and logarithm, even and odd functions, and trigonometric functions (sine, cosine, tangent, arcsine, arctangent) —, with their domain, range, growth or symmetry, and several recognition challenges in each room.
- Room 6: a bridge room that brings the families together in eight graphs and asks to classify them before the test.
- Rooms 7–8: a cross-family recognition test — from graph to formula (6 options) and from formula to graph (4 options).
- Rooms 9–14: the sketching method (domain, axis intercepts, sign, growth and extrema using f′, extra points, behavior at infinity) explained once and applied to each family on empty axes, placing the points found on the plane, and closing with a less guided integrative challenge on a randomly chosen function.
- Rooms 15–21: limits and discontinuities — the limit at a point and the continuity of a piecewise function, jump discontinuity (with one-sided continuity), infinite discontinuity (vertical asymptote), removable discontinuity (hole) with a limit computed by factoring, limits at infinity of quotients of polynomials, a discontinuity-classification challenge ending with a case that has no graph, and a final reading of a graph without a formula that gathers every phenomenon.
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.
- A function is even if f(−x) = f(x) (graph symmetric about the y-axis) and odd if f(−x) = −f(x) (symmetric about the origin); many functions are neither.
- 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, but f′ being zero is not enough: it has to change sign.
- The limit of a function at a point is the value it approaches from both sides; if the function is continuous, substituting is enough. If the one-sided limits are finite but 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. The numerator and denominator vanishing together does not settle the type: you have to simplify and see whether a vanishing factor is left in the denominator.
- Limits at infinity describe what a function does as x grows or shrinks without bound: a quotient of polynomials of the same degree settles at the ratio of the leading coefficients (a horizontal asymptote), and if the top degree is larger, it goes off to infinity.
Room-by-room contents
Room 1 · Families: polynomial functions
A review of quadratics and cubics with two annotated graphs: f(x) = 0.55x^2 − 2 (domain ℝ, range [−2, ∞), a single minimum) and f(x) = 0.3x^3 − 1 (domain and range ℝ, always increasing; other cubics have a decreasing stretch between two extrema). Then six multiple-choice challenges on two new graphs: recognize the family of a downward-opening parabola, its range ((−∞, 3]) and its domain (ℝ); recognize a cubic and its range (ℝ); and, to finish, the x-intercepts of −0.6x^2 + 3 (x = ±√5). Each wrong answer comes with its own hint.
Student tasks
- Recognize a quadratic and give its range ((−∞, 3]) and domain (ℝ).
- Recognize a cubic and give its range (ℝ).
- Find the x-intercepts of −0.6x^2 + 3 (x = ±√5).
Room 2 · Families: rational functions
The hyperbola y = 4/x (k positive: quadrants I and III; domain ℝ\{0}, with the pole at the origin) and its shifted version (ax + b)/(cx + d), with the zero at x = −b/a, the vertical asymptote at x = −d/c and the horizontal one at y = a/c. Challenges on f(x) = (2x + 6)/(4x − 8): domain (ℝ\{2}), vertical asymptote (x = 2) and horizontal asymptote (y = 0.5). Then, recognize the equation of a hyperbola in quadrants II and IV (−12/x) and choose, among four rational functions, the one with horizontal asymptote y = 3 ((3x + 1)/(x − 2)). The hints separate the zero from the vertical asymptote and recall that a vertical asymptote is written x = … and a horizontal one y = ….
Student tasks
- Give the domain and asymptotes of (2x + 6)/(4x − 8) (ℝ\{2}, x = 2, y = 0.5).
- Choose the equation of a hyperbola in quadrants II and IV (−12/x).
- Choose the rational function with horizontal asymptote y = 3.
Room 3 · Families: exponential and logarithm
The exponential 2^x (domain ℝ, range (0, ∞), passes through (0, 1), horizontal asymptote y = 0) and the logarithm log₂(x), its inverse (domain (0, ∞), range ℝ, passes through (1, 0), vertical asymptote x = 0); the base decides whether they increase or decrease. Six challenges: recognize an exponential (1.5^x) and give its domain (ℝ) and range ((0, ∞)); recognize a logarithm (base 3) and its domain ((0, ∞)); and the domain of log₂(x − 3), which is (3, ∞).
Student tasks
- Recognize the exponential and give its domain and range.
- Recognize the logarithm and give its domain.
- Give the domain of log₂(x − 3) ((3, ∞)).
Room 4 · Even and odd functions
The two definitions with an example each: even if f(−x) = f(x), with a graph symmetric about the y-axis (x^2); odd if f(−x) = −f(x), symmetric about the origin (x^3). Challenges: mark, among four graphs, all the even ones (x^2 and cos x) and all the odd ones (x^3 and sin x); decide what the product of an even and an odd function is (odd); classify x^3 + 1 (neither even nor odd); and work out f(0) if f is odd and defined at 0 (f(0) = 0).
Student tasks
- Mark the even graphs (x^2, cos x) and the odd ones (x^3, sin x).
- Decide what the product of an even and an odd function is (odd).
- Classify x^3 + 1 (neither even nor odd).
- Work out f(0) for an odd function defined at 0 (0).
Room 5 · Families: trigonometric functions
Five annotated graphs: sine and cosine (bounded between −1 and 1, period 2π; sine is odd and cosine is even), tangent (period π, vertical asymptotes at π/2 + kπ), arcsine (domain [−1, 1]) and arctangent (domain ℝ, range (−π/2, π/2)). Challenges: recognize sine, cosine and tangent from their graphs; decide that sine is odd; choose which one gets ever closer to a value without reaching it (arctangent); and find the period of sin(3x) (2π/3).
Student tasks
- Recognize sine, cosine and tangent from their graphs.
- Decide whether sine is even or odd (odd).
- Choose the function that approaches a value without reaching it (arctangent).
- Find the period of sin(3x) (2π/3).
Room 6 · Recognize the family
Bridge room before the test: eight graphs, one from each family (quadratic, cubic, rational, exponential, logarithmic, sine, cosine and tangent), in random order and with variants. For each one the family is chosen among six options; when the answer is sine, cosine always appears among the options, and vice versa, and the hint looks at whether the curve passes through the origin. A synthesis challenge closes the room: which family can have both a vertical and a horizontal asymptote (rational).
Student tasks
- Recognize the family of eight graphs.
- Choose the family that can have both a vertical and a horizontal asymptote (rational).
Room 7 · Test: from graph to formula
Eight rounds drawn at random from a bank of thirteen functions: a graph is shown and the formula is chosen among six. The distractors come from other families or from the same family with one change (the sign, the base, cotangent instead of tangent…), and each has its own hint. In the final round every option is the same parabola shifted, so recognising the family is not enough: the vertex has to be read to choose −(x − 1)^2 + 2.
Student tasks
- Choose the formula for eight graphs.
- Read the vertex of a shifted parabola and choose −(x − 1)^2 + 2.
Room 8 · Test: from formula to graph
The reverse of the previous room: the formula is given and the graph is chosen among four, in eight random rounds. If the formula is trigonometric, other trigonometric graphs appear among the options, so that seeing a wavy curve is not enough. After three misses in a round the answer is shown. The final round is 1/(x − 2) + 1 against three shifts of the same hyperbola, each with its own hint.
Student tasks
- Choose the graph for eight formulas.
- Pick out 1/(x − 2) + 1 among shifts of the same hyperbola.
Room 9 · The method for sketching a function
The six steps of the method, one per screen, on f(x) = x^3 − 3x: domain, intercepts with the axes (0 and ±√3), sign, growth and extrema using f′ (maximum at (−1, 2) and minimum at (1, −2)), a few extra points and behavior at infinity; the graph marks what is found at each step. Then three challenges: in which step the derivative is used (step 4), where the derivative of g(x) = x^3 − 12x is zero (x = ±2), and whether f′(a) = 0 guarantees a maximum or a minimum (no: f′ has to change sign, as x^3 at 0 shows).
Student tasks
- Go through the six steps of the method on x^3 − 3x.
- Say in which step the derivative is used (step 4).
- Find where the derivative of x^3 − 12x is zero (x = ±2).
- Decide whether f′(a) = 0 guarantees an extremum (not always).
Room 10 · Sketch: a polynomial function
The method applied to g(x) = x^3 − 3x^2 on empty axes: the curve is not drawn until the end. The student answers in order the domain (ℝ) and the y-intercept ((0, 0)), which they then place with a click on the plane; marks the x-intercepts among several candidates (x = 0 and x = 3); finds where g′(x) = 3x^2 − 6x is zero (x = 0 and x = 2) and decides what is at x = 2 (a minimum); calculates g(2) = −4 and places that point. At the end the full graph is drawn with everything found marked on it.
Student tasks
- Give the domain and the y-intercept, and place it on the plane.
- Mark the x-intercepts (x = 0 and x = 3).
- Find the zeros of g′ (x = 0 and x = 2) and classify x = 2 (minimum).
- Calculate g(2) (−4) and place the minimum.
Room 11 · Sketch: a rational function
The method applied to h(x) = (−x − 1)/(x + 2), also on empty axes: domain (ℝ\{−2}), x-intercept ((−1, 0), which is placed on the plane), horizontal asymptote (y = −1) and where h shoots off just to the right of the pole (+∞), looking at the signs of the numerator and the denominator separately. At the end the full graph appears.
Student tasks
- Give the domain (ℝ\{−2}).
- Find and place the x-intercept ((−1, 0)).
- Give the horizontal asymptote (y = −1).
- Decide where h goes just to the right of x = −2 (+∞).
Room 12 · Sketch: exponential and logarithm
The method applied to k(x) = 2^x − 1: domain (ℝ), y-intercept ((0, 0), which is placed on the plane), what k approaches as x → −∞ (−1, horizontal asymptote) and whether it has a maximum or minimum (no, it is increasing: the derivative of an exponential is never zero). With no extrema, it is the asymptote that gives the graph its shape.
Student tasks
- Give the domain and place the y-intercept ((0, 0)).
- Give the limit as x → −∞ (−1).
- Decide whether k has extrema (no).
Room 13 · Sketch: a trigonometric function
The method applied to m(x) = sin(x) on [0, 2π]: domain (ℝ), x-intercepts (0, π and 2π, which are placed on the plane), zeros of m′(x) = cos(x) (π/2 and 3π/2) and what is at π/2 (a maximum, which is also placed). The final graph also marks the minimum at 3π/2.
Student tasks
- Give the domain (ℝ).
- Find and place the x-intercepts on [0, 2π] (0, π, 2π).
- Find the zeros of the derivative (π/2 and 3π/2).
- Classify and place the point at x = π/2 (maximum).
Room 14 · Integrative challenge
The full method on a function chosen at random from four, one per family: −x^3 + 3x, −2/(x − 1), 3^x − 4 or cos(x). There are three questions per function, less guided than in the previous rooms (domain or range, intercepts, zeros of the derivative, asymptotes or behavior next to the pole); in three of the four there is also a point to place on the plane. At the end the full graph is drawn.
Student tasks
- Apply the method to the function that comes up.
- Place on the plane the points asked for (an extremum or the intercepts).
Room 15 · Limit at a point: approach from both sides
Two sliders, one red and one green, move a point towards x = 1 from the left and another from the right on f(x) = x^2 − x + 1, showing the value of f at each. When both arrive, the conclusion appears: the two one-sided limits equal f(1), so the function is continuous at x = 1. Challenges: the limit of x^2 − x + 1 as x → 2 (3, by substitution, since it is continuous) and the value of a that makes continuous at x = 1 the function equal to x^2 − x + 1 for x < 1 and 2x + a for x ≥ 1 (a = −1).
Student tasks
- Move both points to x = 1 and compare the one-sided limits with f(1).
- Calculate the limit of x^2 − x + 1 as x → 2 (3).
- Find the value of a that makes the piecewise function continuous (−1).
Room 16 · Jump discontinuity
A piecewise function, x + 1 to the left of 1 and x − 1 to the right, with an open point at (1, 2) and a filled one at (1, 0). Four chained questions: the left-hand limit (2), the right-hand limit (0), whether the limit at x = 1 exists (no, because they differ) and whether f is right-continuous at x = 1 (yes: f(1) equals the right-hand limit). The room ends with the definition: finite but different one-sided limits make a jump discontinuity.
Student tasks
- Read the two one-sided limits at x = 1 (2 and 0).
- Decide whether the limit exists (no).
- Decide whether f is right-continuous at x = 1 (yes).
Room 17 · Infinite discontinuity
f(x) = 1/(x − 2) in two panels with the asymptote drawn: one slider moves a point in from the left and another from the right, and the vertical axis rescales to follow them towards −∞ and +∞. First the student chooses what happens to f(x) near 2 (it shoots off: vertical asymptote); then the left-hand limit (−∞) and which of four functions has an infinite discontinuity at x = 3 (1/(x − 3)^2; of the others, two simplify and leave only a hole, and the third has its pole at −3).
Student tasks
- Move the points in from both sides and describe what happens to f.
- Give the left-hand limit at x = 2 (−∞).
- Choose the function with an infinite discontinuity at x = 3 (1/(x − 3)^2).
Room 18 · Removable discontinuity
f(x) = (x^2 − 4)/(x − 2), with the formula on screen and the two sliders approaching a hole at x = 2. Questions: whether f(2) is defined (no) and whether the limit as x → 2 exists (yes, it is 4, even though f(2) does not exist). Conclusion: defining f(2) = 4 would be enough to fill the hole. A third challenge, by hand: the limit of (x^2 − 3x + 2)/(x − 1) as x → 1, by factoring and simplifying (−1).
Student tasks
- Decide whether f(2) is defined (no).
- Decide whether the limit at x = 2 exists and give its value (4).
- Calculate the limit of (x^2 − 3x + 2)/(x − 1) at x = 1 (−1).
Room 19 · Limits at infinity
f(x) = (2x + 1)/(x − 1) with a reach slider that zooms out from x = ±5 to x = ±400, with the line y = 2 dashed. The student types the value f(x) approaches as x → ±∞ (2) and the name appears: horizontal asymptote. Challenges: the limit of (3x − 5)/(6x + 1) at +∞ (1/2, the ratio of the coefficients of x when top and bottom have the same degree) and that of (x^2 + 1)/(x − 1) (+∞, because the top has the higher degree).
Student tasks
- Zoom out and type what f(x) approaches (2).
- Calculate the limit of (3x − 5)/(6x + 1) at +∞ (1/2).
- Calculate the limit of (x^2 + 1)/(x − 1) at +∞ (+∞).
Room 20 · Classify the discontinuity
Three graphs in random order, a jump, a vertical asymptote and a hole, to be classified as jump, infinite or removable. The last case has no graph: f(x) = (x^2 − 4)/(x^2 − 4x + 4) at x = 2. It looks removable because a factor (x − 2) cancels, but the denominator is (x − 2)^2 and one factor remains below: it is infinite.
Student tasks
- Classify three discontinuities from their graphs.
- Classify, without a graph, that of (x^2 − 4)/(x^2 − 4x + 4) at x = 2 (infinite).
Room 21 · Read the graph: everything at once
A piecewise function drawn with no formula brings the whole block together: vertical asymptote at x = −2, jump at x = 1, hole at x = 4, horizontal asymptote y = 3, relative maximum at x = −8 and relative minimum at x = −5. The student locates each feature by choosing among four values, with no calculation, and finally counts at how many points the function is discontinuous (3: the extrema and the horizontal asymptote do not break the curve).
Student tasks
- Locate the vertical asymptote, the jump and the hole (x = −2, 1 and 4).
- Give the horizontal asymptote (y = 3).
- Locate the relative maximum and minimum (x = −8 and x = −5).
- Count the points of discontinuity (3).
Rooms to project
The most striking ones to show and discuss in class.