Teaching objectives
What are functions? They may be the most important concept in mathematics, and yet few students ever truly understand them. Through real-life examples like exam scores and metaphors like a postman delivering letters, we introduce the ideas of domain, range, and value. Students will learn that a function is a transformation, explore domain and range visually, learn to graph affine functions, parabolas and hyperbolas by following a step-by-step algorithm, and meet the main families taught in secondary school and upper secondary: affine, quadratic, cubic and rational.
What you'll learn
Students build an intuitive understanding of a function as a correspondence, practice reading graphs, discover how parameters shape a curve, and explicitly learn the procedure for graphing each family by hand.
- Room 1: the linear family (y = m·x); the slider m controls the slope.
- Room 2: guided tutorial — graphing an affine function point by point (pick x, compute f(x), place the point), twice (increasing and decreasing).
- Rooms 3–9: the affine family (y = m·x + n) and the parabola in its forms y = x², y = a·x² + c, y = x² + b·x + c and the general y = a·x² + b·x + c, with challenges to "hit a given range", "hit given roots and concavity", and "hit a given axis of symmetry".
- Room 10: guided tutorial — the full algorithm for graphing a parabola (vertex, y-intercept, x-intercepts if any, extra points).
- Rooms 11–12: the cubic family, first y = a·x³ + d (center of symmetry, monotonicity) and then the general y = x³ + b·x² + c·x + d (roots, range, domain).
- Rooms 13–14: formula↔graph recognition across parabolas and cubics.
- Rooms 15–17: the rational family (hyperbolas) y = k/x, y = k/(x−v) + h and y = (a·x+b)/(c·x+d), with "hit these asymptotes" challenges.
- Room 18: guided tutorial — the full algorithm for graphing a hyperbola (asymptotes, axis intercepts).
- Room 19: final recognition crossing all four studied families.
Key mathematical ideas
- A function is a one-to-one correspondence: one x → exactly one f(x). If any x maps to two outputs, it is not a function.
- Domain: the set of x-values for which the function is defined. Image (range): the set of values that f(x) takes.
- Graphing by hand always follows the same pattern: pick strategic points (vertex, axis intercepts, asymptotes), compute their value by substituting into the formula, and place them on the plane before joining them with the curve.
- The linear function y = m·x passes through the origin; m determines whether it increases (m > 0) or decreases (m < 0).
- In the parabola y = a·x² + c, the sign of a decides whether it opens upward or downward (minimum/maximum), and the number of roots depends on the position of the vertex relative to the x-axis.
- The vertex of y = ax² + bx + c is located at x = −b/(2a); the axis of symmetry is the vertical line through that point.
- A cubic y = a·x³ + d is always monotonic and has a center of symmetry at (0, d); being odd-degree, any cubic always has at least one real root, and its domain and range are always ℝ.
- A hyperbola y = (a·x+b)/(c·x+d) has a vertical asymptote at x = −d/c (where the denominator vanishes) and a horizontal asymptote at y = a/c (the ratio of the leading coefficients); neither the domain nor the range includes those values.
Room-by-room contents
Room 1 · The linear function y = m·x
The linear family y = m·x is explored with a slider for m. The line and the value table update live; students observe when the function increases or decreases depending on the sign of m.
Student tasks
- Move m and describe what changes in the line (slope, direction).
- State for which values of m the function is increasing and for which it is decreasing.
Room 2 · Tutorial: graphing an affine function
A step-by-step guided tutorial. First the lab itself models working out the first point (pick an x, substitute into f(x), compute, fill in the table, plot the point). Then the student repeats the process for two more points: tap the marked x on the axis, compute f(x) by typing the result, and place the correct point on the graph. With 3 points the line appears. The whole process repeats twice: once for an increasing function, once for a decreasing one.
Student tasks
- Follow the demonstration of the first point.
- Complete the other two points yourself: tap the axis, compute f(x), and mark the point on the graph.
- Repeat the whole process for a decreasing function.
Room 3 · The affine function y = m·x + n
The parameter n is added to the previous function. With two sliders (m and n), the student discovers the effect of each one and identifies the domain and range of the affine function.
Student tasks
- Manipulate m and n separately and describe what each parameter changes.
- Answer: what are the domain and range of y = m·x + n?
Room 4 · The parabola y = x²
The basic quadratic function is presented through an animated table that reveals the curve. The vertex (0, 0) and the axis of symmetry x = 0 are highlighted, and the growth is contrasted with that of a line.
Student tasks
- Calculate or verify several values in the table and observe the U shape.
- Locate the vertex and axis of symmetry on the graph.
Room 5 · The parabola y = a·x² + c
Two sliders (a and c) allow the parabola to be scaled and shifted vertically. The student observes how the sign of a determines concavity and range, and how the combination of signs affects the number of real roots.
Student tasks
- Explore different sign combinations of a and c and count the roots in each case.
- Describe how the range of the function changes as a and c vary.
Room 6 · Hit the range
A practical challenge: three range targets are proposed and the student must adjust the a and c sliders of y = a·x² + c until the range of their parabola matches exactly what is asked.
Student tasks
- Adjust a and c so that the range of the parabola matches each proposed target.
- Complete all 3 range challenges.
Room 7 · Hit the roots and concavity
Continuing with y = a·x² + c, three combinations of number of roots (0, 1 or 2) and type of extremum (maximum or minimum) are posed. The student chooses the correct signs to meet each condition.
Student tasks
- Determine which signs of a and c produce each required combination of roots and extremum.
- Complete all 3 configuration challenges.
Room 8 · Parabola y = x² + b·x + c
The term b·x is added. With sliders b and c, the vertex, the axis of symmetry (x = −b/2) and the roots (when they exist) are shown live. Also includes two challenges: hit a given axis of symmetry, and hit a given axis of symmetry together with a given minimum.
Student tasks
- Move b and c and record how the vertex shifts and the axis of symmetry changes.
- Find configurations with 0, 1 and 2 real roots.
- Complete the 2 axis-of-symmetry challenges.
Room 9 · General parabola y = a·x² + b·x + c
With all three parameters active, a complete analysis is carried out: concavity (sign of a), vertex, axis of symmetry, range and roots. Includes two extra challenges with a negative: hit a given axis of symmetry, plus a given maximum.
Student tasks
- Explore all three sliders (a, b, c) and describe the effect of each.
- For a chosen parabola, identify the vertex, axis of symmetry, range and number of roots.
- Complete the 2 challenges with a negative.
Room 10 · Tutorial: graphing a parabola
A guided tutorial covering the full algorithm: compute the discriminant and the vertex (x_v = −b/2a, y_v = f(x_v)), the y-intercept, the roots (in any order), and add a couple of extra table points. Every computed value is placed on the graph by tapping the correct point. Repeated across 2 exercises with real roots (a>0 and a<0).
Student tasks
- Compute the vertex of the parabola and mark it on the graph.
- Compute the discriminant and both roots.
- Complete both guided exercises.
Room 11 · The cubic y = a·x³ + d
Two sliders (a and d) show how this cubic family is always monotonic (no local maxima or minima) and has a center of symmetry at (0, d). A short quiz reviews the center of symmetry, the range, and the role of each parameter.
Student tasks
- Move a and d and observe how the curve changes.
- Answer the quiz about center of symmetry, range and monotonicity.
Room 12 · The general cubic y = x³ + b·x² + c·x + d
With three sliders (b, c, d) the student freely explores the general cubic. The quiz stresses that, being odd-degree, it always has at least one real root, and that its domain and range are always ℝ, with answer options designed to tempt confusion with the parabola case.
Student tasks
- Move b, c and d and observe the different curve shapes that appear.
- Answer the quiz about roots, range and domain.
Room 13 · Recognize the graph
A formula is shown (parabola or cubic, with positive or negative leading coefficient) and the student picks which of 4 mini-graphs matches it. Repeated 3 times with different formulas and positions each time.
Student tasks
- Read the given formula and identify its family and the sign of its leading coefficient.
- Complete all 3 recognition challenges.
Room 14 · Recognize the formula
The reverse of the previous room: a graph is shown and the student picks which of 4 formulas matches it. Repeated 3 times.
Student tasks
- Look at the given graph and identify its family and the sign of its leading coefficient.
- Complete all 3 recognition challenges.
Room 15 · The hyperbola y = k/x
A single slider for k presents the simplest hyperbola. The quiz covers the asymptotes (always x=0 and y=0), the domain and range (both ℝ\{0}), which quadrants the curve lives in depending on the sign of k, and the nuance that it decreases on each branch separately even though it isn't monotonic over its whole domain.
Student tasks
- Move k and observe how the two branches of the curve change.
- Answer the quiz about asymptotes, domain, range and per-branch monotonicity.
Room 16 · The translated hyperbola y = k/(x−v) + h
With three sliders (k, v, h) the asymptotes shift to x=v (vertical) and y=h (horizontal). The quiz asks students to read the asymptote positions directly from v and h and deduce the domain and range.
Student tasks
- Move v and h and observe how the asymptotes shift along with the curve.
- Answer the quiz about asymptotes, domain and range.
Room 17 · General hyperbola y = (a·x+b)/(c·x+d)
With four sliders (a, b, c, d) the student must hit two challenges: a specific vertical asymptote (x = −d/c) and, in the second challenge, a vertical and a horizontal asymptote at the same time (x = −d/c and y = a/c).
Student tasks
- Work out which values of c and d give the requested vertical asymptote.
- Also work out which values of a and c give the requested horizontal asymptote.
- Complete both asymptote challenges.
Room 18 · Tutorial: graphing a hyperbola
A guided tutorial covering the full algorithm for a hyperbola: vertical asymptote (solve c·x+d=0), horizontal asymptote (a/c), y-intercept and x-intercept. Every computed value is placed on the graph. Repeated across 2 exercises with different hyperbolas.
Student tasks
- Compute both asymptotes of the given hyperbola.
- Compute the axis intercepts and mark them on the graph.
- Complete both guided exercises.
Room 19 · Final recognition
Lab closing: a formula from any of the studied families (affine, parabola, cubic or hyperbola) is shown and the student picks which of 6 mini-graphs matches it. Repeated 3 times, mixing different families each round.
Student tasks
- Recognize the family of the given formula among the 6 proposed graphs.
- Complete all 3 final challenges.
Rooms to project
The most striking ones to show and discuss in class.