Graph_of_a_function - Wikiparency

# Graph of a function ## Microsim (three.js) <div class="microsim-player">  <p class="wt-pending"><strong>Microsim staged, not yet on the CDN.</strong> <code>Graph_of_a_function.html</code> is built and deploy-ready in <code>Microsims for Dissemination/</code>, but the Netlify project still serves the geometry+spintronics set only. The player is disabled until the deploy lands; the explanatory text below is unchanged.</p>   </div> *Every plotted curve is a small algorithm: sample the domain, evaluate $f$ in Floating-point arithmetic, and draw the result. Formally the graph is nothing but a [[Set_theory|set]] of ordered pairs — and the very surfaces a [[Graphics_processing_unit]] shades are where an optimizer hunts for the peaks and valleys promised by the Extreme value theorem.* > The graph of a function is the picture you get by plotting every input beside its output. For a rule like $f(x)=x^2$, each number $x$ is paired with a height $f(x)$, and the collection of all such points forms a curve. This microsim lets you pick a rule — Linear, Quadratic, Sine, or Exponential — and slide $x$ from −5 to 5 to watch the matching point trace that curve. Switch to Surface mode and the same idea gains a dimension: a function of two inputs becomes a landscape floating above the floor. ## About this microsim Drag the **x** slider (`x = 0.00`, running from −5 to 5) and a marker rides along the horizontal axis while its partner coordinate climbs or dips to meet the curve, so you literally see the pair $(x, f(x))$ the definition names. Tap **Linear**, **Quadratic**, **Sine**, or **Exponential** to swap the rule and watch the same slider trace a straight line, a parabola, a wave, or a runaway climb — the shape *is* the function. Choosing **Surface z=f(x,y)** lifts the construction into three dimensions, replacing the curve with a surface you can orbit; a graph, it shows, is not tied to the plane but is simply the set of input–output points, however many inputs there are. ## Related microsims - [[Set_theory]] — a graph is literally a set of ordered pairs $(x, f(x))$; its formal definition lives here. - Extreme value theorem — the peaks and valleys you see are the guaranteed maxima and minima of the curve. - Floating-point arithmetic — every plotted coordinate is a float; sampling and rounding shape what appears. - [[Graphics_processing_unit]] — the $z=f(x,y)$ surface is shaded by the same pipeline a GPU runs. - Raster graphics — turning a continuous curve into pixels is a sampling and rasterisation problem. - [[Calculus_of_variations]] — when the input is itself a function, you optimize over a whole space of graphs. ## Links (Wikipedia order)  `Aaron_Koblin` · `Abel's_test` · `Adequality` · `Adolphe_Quetelet` · `Alan_MacEachren` · `Alfred_Inselberg` · `Algebraic_function` · `Alternating_series` · `Alternating_series_test` · `Analytic_function` · `André-Michel_Guerry` · `Antiderivative` · `Arc_length` · `Arithmetico-geometric_sequence` · `Arthur_H._Robinson` · `Arthur_Lyon_Bowley` · `Asymptote` · `August_Kekulé` · `Bang_Wong` · `Ben_Fry` · `Ben_Shneiderman` · `Bernoulli_number` · `Bijection` · `Binary_relation` · `Binomial_series` · `Binomial_theorem` · `Biological_data_visualization` · `Boolean-valued_function` · `Boolean_function` · `Borden_Dent` · `Brook_Taylor` · `Bruce_H._McCormick` · `CPK_coloring` · [[Calculus]] · [[Cartesian_coordinate_system]] · `Cartesian_product` · `Cartography` · `Catherine_Plaisant` · `Cauchy_condensation_test` · `Chain_rule` · `Charles-René_de_Fourcroy` · `Charles_Booth_(social_reformer)` · `Charles_Dupin` · `Charles_Joseph_Minard` · `Chart` · `Chartjunk` · `Chemical_imaging` · `Christopher_R._Johnson` · `Claudio_Silva_(computer_scientist)` · `Clifford_A._Pickover` · `Codomain` · `Colin_Maclaurin` · `Computer_graphics` · `Computer_graphics_(computer_science)` · `Concave_function` · `Constant_function` · `Constant_of_integration` · `Continuous_function` · `Contour_line` · `Convex_function` · `Crime_mapping` · `Critical_point_(mathematics)` · `Curl_(mathematics)` · `Curvature` · `Cynthia_Brewer` · `Daniel_A._Keim` · `David_Goodsell` · `David_McCandless` · `Derivative` · `Diagram` · `Differential_(mathematics)` · `Differential_calculus` · [[Differential_equation]] · `Differential_form` · `Differential_geometry_of_surfaces` · `Differential_operator` · `Differentiation_rules` · `Direct_comparison_test` · `Directional_derivative` · `Dirichlet's_test` · `Disc_integration` · `Divergence` · `Divergence_theorem` · `Domain_of_a_function` · `E_(mathematical_constant)` · `Edgar_Anderson` · `Edmond_Halley` · `Edward_Tufte` · `Edward_Walter_Maunder` · `Ejnar_Hertzsprung` · [[Engineering]] · [[Engineering_drawing]] · `Epigraph_(mathematics)` · `Erwin_Raisz` · `Euler_substitution` · `Euler–Maclaurin_formula` · `Exponential_function` · `Exterior_derivative` · `Extreme_value_theorem` · `Factorial` · `Fernanda_Viégas` · `Finance` · `Finite_difference` · `Florence_Nightingale` · `Flow_visualization` · `Fluxion` · `Foundations_of_mathematics` · `Fourier_series` · `Francis_Amasa_Walker` · `Francis_Galton` · `Free_variables_and_bound_variables` · `Fritz_Kahn` · `Function_(mathematics)` · `Function_composition` · `Function_of_a_real_variable` · `Function_of_several_complex_variables` · `Function_of_several_real_variables` · `Function_space` · `Functor` · `Fundamental_theorem_of_calculus` · `Gabriel's_horn` · `Gaspard_Monge` · `General_Leibniz_rule` · `Generality_of_algebra` · `Geometric_calculus` · `Geometric_series` · `George_Furnas` · `George_G._Robertson` · `Geovisualization` · `Gordon_Kindlmann` · `Gottfried_Wilhelm_Leibniz` · `Gradient` · `Graph_drawing` · `Graphic_design` · `Graphic_organizer` · `Green's_theorem` · `Hadley_Wickham` · `Hans_Rosling` · `Hanspeter_Pfister` · `Harmonic_series_(mathematics)` · `Harry_Beck` · `Henry_Gantt` · `Henry_Norris_Russell` · `Hessian_matrix` · `Higher-order_function` · `History_of_calculus` · `History_of_the_function_concept` · `Howard_G._Funkhouser` · `Howard_T._Fisher` · `Howard_Wainer` · `Identity_function` · `Ideogram` · `Imaging` · `Implicit_differentiation` · `Implicit_function` · `Indeterminate_form` · `Infinitesimal` · `Infographic` · `Information_art` · [[Information_science]] · `Injective_function` · `Integer-valued_function` · `Integral` · `Integral_equation` · `Integral_of_secant_cubed` · `Integral_of_the_secant_function` · `Integral_test_for_convergence` · `Integration_Bee` · `Integration_by_parts` · `Integration_by_substitution` · `Integro-differential_equation` · `Internet_Archive` · `Interval_(mathematics)` · `Inverse_function` · `Inverse_function_rule` · [[Isaac_Newton]] · `Jacobian_matrix_and_determinant` · `Jacques_Bertin` · `Jeffrey_Heer` · `Jessica_Hullman` · `Jock_D._Mackinlay` · `John_B._Peddle` · `John_Maeda` · `John_Snow` · `John_Tukey` · `John_Venn` · `Joseph_Priestley` · `Karl_Wilhelm_Pohlke` · `Kwan-Liu_Ma` · `L'Hôpital's_rule` · `Lagrange_multiplier` · `Lambda_calculus` · `Laplace_operator` · `Lawrence_J._Rosenblum` · `Leibniz's_notation` · `Leibniz_integral_rule` · `Leland_Wilkinson` · `Leonhard_Euler` · `Limit_(mathematics)` · `Limit_comparison_test` · `Limit_of_a_function` · `Limit_of_a_sequence` · `Line_integral` · `Linear_function` · `Linear_map` · `Linearity_of_differentiation` · `List_of_integrals_of_exponential_functions` · `List_of_integrals_of_hyperbolic_functions` · `List_of_integrals_of_inverse_hyperbolic_functions` · `List_of_integrals_of_inverse_trigonometric_functions` · `List_of_integrals_of_irrational_algebraic_functions` · `List_of_integrals_of_logarithmic_functions` · `List_of_integrals_of_rational_functions` · `List_of_integrals_of_trigonometric_functions` · `List_of_limits` · `List_of_mathematical_functions` · `List_of_types_of_functions` · `Lists_of_integrals` · `Logarithmic_derivative` · `Manifold` · `Manuel_Lima` · `Map` · `Map_(mathematics)` · `Martin_M._Wattenberg` · `Mary_Eleanor_Spear` · `Mathematical_diagram` · `Mathematics` · `Matrix_calculus` · `Matthew_Henry_Phineas_Riall_Sankey` · `Mauro_Martino` · `Max_O._Lorenz` · `Maximum_and_minimum` · `Mean_value_theorem` · `Measurable_function` · `Medical_imaging` · `Method_of_Fluxions` · `Michael_Friendly` · `Miriah_Meyer` · `Misleading_graph` · `Molecular_graphics` · `Moritz_Stefaner` · `Morphism` · `Multiple_integral` · `Multivalued_function` · `Multivariable_calculus` · `Natural_logarithm` · `Neuroimaging` · `Newton's_method` · `Nigel_Holmes` · `Normal_(geometry)` · `Notation_for_differentiation` · `Oliver_Byrne_(mathematician)` · `One-sided_limit` · `Order_of_approximation` · `Ordered_pair` · [[Ordinary_differential_equation]] · `Otto_Neurath` · `Partial_derivative` · [[Partial_differential_equation]] · `Partial_function` · `Pat_Hanrahan` · `Patent_drawing` · `Photograph` · `Pictogram` · `Plane_curve` · `Plot_(graphics)` · `Polynomial` · `Power_rule` · `Power_series` · `Precalculus` · `Product_rule` · `Proof_that_22/7_exceeds_π` · `Quadratic_integral` · `Quotient_rule` · `Radian` · `Range_of_a_function` · `Ratio_test` · `Rational_function` · `Real-valued_function` · `Real_number` · `Regiomontanus'_angle_maximization_problem` · `Related_rates` · `Relation_(mathematics)` · `Restriction_(mathematics)` · `Riemann_integral` · `Rolle's_theorem` · `Root_test` · `Rudolf_Modley` · `Sankey_diagram` · `Schematic` · [[Science]] · `Scientific_modelling` · `Scientific_visualization` · `Secant_line` · `Second_derivative` · [[Sequence]] · `Series_(mathematics)` · `Set-valued_function` · [[Set_theory]] · `Sheelagh_Carpendale` · `Shell_integration` · `Skeletal_formula` · `Slope` · `Social_visualization` · `Software_visualization` · `Spatial_analysis` · `Stationary_point` · `Statistical_graphics` · `Steinmetz_solid` · `Stirling's_approximation` · `Stochastic_differential_equation` · `Stokes'_theorem` · `Stuart_Card` · `Surface_(mathematics)` · `Surface_integral` · `Surjective_function` · `Table_(information)` · `Tamara_Munzner` · `Tangent` · `Tangent_half-angle_substitution` · `Taylor's_theorem` · `Taylor_series` · `Technical_drawing` · `Technical_illustration` · `Technology` · `Telescoping_series` · [[Tensor]] · `Tetraview` · `The_Method_of_Mechanical_Theorems` · `Thomas_A._DeFanti` · `Three-dimensional_space` · `Tom_M._Apostol` · `Toussaint_Loua` · `Trapezoidal_rule` · `Trigonometric_substitution` · `User_interface_design` · `Vector_calculus` · `Virtual_unfolding` · `Visual_analytics` · `Visual_culture` · `Visual_perception` · `Visualization_(graphics)` · `Volume_integral` · `Volume_rendering` · `W._E._B._Du_Bois` · `William_Playfair` · `William_S._Cleveland` · `Y-intercept` ## Overview In mathematics, the **graph of a function** $f$ is the set of all ordered pairs $(x, f(x))$ that pair each input with its output. When inputs and outputs are real numbers, those pairs are Cartesian coordinates and the graph appears as a curve in the plane — an object drawn since René Descartes and Pierre de Fermat fused algebra with geometry in the seventeenth century. The graph turns an abstract rule into something visible: slopes, peaks, zeros, and asymptotes all become features you can point at. In computing it underpins data plotting, function visualisation, machine-learning loss "landscapes," and surface rendering. ## The mathematics Given a function $f : X \to Y$, its graph is the subset of the Cartesian product $G(f) = \{\,(x, f(x)) : x \in X\,\} \subseteq X \times Y.$ For a single real variable this is a curve in $\mathbb{R}^2$; for two variables, $f:\mathbb{R}^2\to\mathbb{R}$, it is a surface in $\mathbb{R}^3$, namely $\{(x,y,f(x,y))\}$ — exactly the Surface mode. A set of points is the graph of a function precisely when it passes the **vertical line test**: no input maps to two outputs. The four built-in rules span a range of shapes: | Mode | Formula | Graph | |---|---|---| | Linear | $f(x)=x$ | straight line, constant slope | | Quadratic | $f(x)=x^2$ | parabola, single minimum at $0$ | | Sine | $f(x)=\sin x$ | periodic wave, period $2\pi$ | | Exponential | $f(x)=e^{x}$ | rapid growth, asymptote as $x\to-\infty$ | For example, in Quadratic mode setting the slider to $x=2$ marks the point $(2,4)$, since $2^2 = 4$. ## Controls -> what each maps to | Control | Maps to | Range / values | Meaning | |---|---|---|---| | `x = 0.00` | input (domain) coordinate | −5 to 5 | Slides a probe point along the x-axis; the marker reads off the pair $(x, f(x))$ | | Linear | selects $f(x)=x$ | mode (on/off) | Straight-line graph | | Quadratic | selects $f(x)=x^2$ | mode (on/off) | Parabola | | Sine | selects $f(x)=\sin x$ | mode (on/off) | Periodic wave | | Exponential | selects $f(x)=e^x$ | mode (on/off) | Exponential curve | | Surface z=f(x,y) | switches to a two-input graph | mode (on/off) | Renders the graph as a 3D surface | ## Learning objective After playing, a learner can predict a graph's shape from its formula, read the point $(x, f(x))$ for any chosen input, and explain why a one-input function draws a curve while a two-input function draws a surface. ## Limits and connections A rendered graph is only a sampled approximation: the sim evaluates $f$ at finitely many points and connects them, so detail between samples can be missed — the same aliasing trade-off faced in Raster graphics. Reading a graph's lowest or highest point is the visual form of optimization, and treating the input itself as a function leads to the [[Calculus_of_variations]]. ## Poster & source <div class="microsim-fallback">  <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/Graph_of_a_function.html">open full</a> · source: Microsims for Dissemination/ALGORITHM_microsims/Graph_of_a_function.html</em></p> </div>  *Built to the [[WT!Three_js_Microsim_Master_Class|three.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Graph_of_a_function) : [Wikitube](https://en.wikitube.io/wiki/Graph_of_a_function) ## Previous hub tags Tree parents: [[Dynamical_system]] · [[Graph_theory]]. Legacy hubs: `ALGORITHM`. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*