Functions & Graphs

• Graph transformations – translations, stretches and reflections of y = a·f(b(x − h)) + k Mathematics · Functions & Graphs Pick a parent function (x, x², x³, |x|, √x, 1/x, 1/x², 2ˣ, eˣ, ln x, sin x) and drag a, b, h and k in g(x) = a·f(b(x... Ages 15–16
• Quantities changing together – sketching graphs for filling vases and a bike ride Mathematics · Functions & Graphs Fill a cylinder, a cone, a round flask, a bowl or a vase you design at a constant rate, or follow a bike ride; drag... Ages 15–16
• Graphing lines – slope and y-intercept Mathematics · Functions & Graphs Four screens for exploring the straight line y = ax + b: drag two points to find the slope with a rise-over-run... Ages 13–14
• Graphing quadratics – coefficients, vertex form, focus and directrix Mathematics · Functions & Graphs Five screens: Explore (drag the a, b, c sliders and see each term ax², bx and c separately), Standard Form (integer... Ages 15–16
• Rational functions – asymptotes of the graph Mathematics · Functions & Graphs For y = (ax + b)/(cx + d) or y = (ax² + bx + c)/(x + d), drag the coefficients to see the vertical, horizontal and... Ages 17–18
• Solving equations and inequalities graphically Mathematics · Functions & Graphs Type any two functions f and g (or a horizontal line y = k) and see the solutions of f(x) = g(x) as the... Ages 15–16
• Limits of Sequences and Functions, Continuity Mathematics · Functions & Graphs Sequence mode: the points (n, uₙ) approach the line y = L; a draggable band L ± ε shows that from some index N₀... Ages 16–17
• Composite and inverse functions – mapping diagrams, graphs and reflection in y = x Mathematics · Functions & Graphs Choose two functions f and g (linear, quadratic, exponential, square root, reciprocal) with parameters a and b. The... Ages 15–16
• Exponential function y = aˣ and logarithmic function y = logₐx Mathematics · Functions & Graphs The graphs of y = aˣ and y = logₐx are reflections of each other across the line y = x. Drag point M on the... Ages 16–17
• Linear functions – graphs and the equation ax + b = 0 Mathematics · Functions & Graphs Graph of y = ax + b on the coordinate plane. Drag the point on the y-axis to change b, drag the second point to... Ages 13–14
• Continuity of a function at a point Mathematics · Functions & Graphs A piecewise function joined at x₀, with an adjustable parameter m. The simulation computes the left-hand limit, the... Ages 16–17
• Parent functions – domain, range, symmetry, asymptotes and piecewise functions Mathematics · Functions & Graphs Explore the graphs and exact key features of the parent functions y = x, x², x³, √x, ∛x, |x|, 1/x, 1/x², xⁿ for any... Ages 15–16
• Cubic and rational function analysis – derivatives, extrema, asymptotes Mathematics · Functions & Graphs Plots y = ax³ + bx² + cx + d or y = (ax + b)/(cx + d) alongside a variation table built live from the sign of y'.... Ages 17–18
• Function model fitting – residuals, R² and linearisation Mathematics · Functions & Graphs Pick a real-context data set (a cooling drink, drug concentration, world population, yeast growth, a thrown ball,... Ages 16–17
• Function machines – inputs, outputs and rules Mathematics · Functions & Graphs Chain up to 3 "machines" in a row to turn an input into an output. Four screens: Patterns (machines that rotate,... Ages 13–14
• Quadratic function y = ax² + bx + c: vertex, axis of symmetry, variation table Mathematics · Functions & Graphs The parabola y = ax² + bx + c is graphed with three sliders for a, b, c right in the simulation; students can also... Ages 15–16
• Parabola y = ax² and quadratic equations Mathematics · Functions & Graphs The graph of y = ax² together with the line y = bx + c. Drag the point on the parabola to change a (opens up/down,... Ages 14–15
• Relations and functions – Cartesian product, arrow diagrams, one-one and onto Mathematics · Functions & Graphs Choose two small sets A and B and see A × B as a lattice of ordered pairs; tap pairs (or draw arrows) to build a... Ages 15–16
• Linear, quadratic or exponential? – Tables, differences, ratios and comparing functions Mathematics · Functions & Graphs See a function as a table, a graph and a formula side by side. Constant first differences Δy mean linear, constant... Ages 15–16
• Log scales and linearisation – semi-log and log–log graphs Mathematics · Functions & Graphs Plot real data on ordinary axes, semi-log axes (log y against x) and log–log axes (log y against log x): y = a·bˣ... Ages 16–17
• Polar coordinates and polar graphs – conversion, roses, limaçons, spirals and area Mathematics · Functions & Graphs Drag a point P to convert between polar coordinates (r, θ) and Cartesian coordinates (x, y), see the other names of... Ages 16–17

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