Function Grapher
Plot y = f(x) · Zoom & Pan · Trace · Roots · Value Table
Enter any function of x — polynomial, trigonometric, logarithmic, or exponential — and see it graphed instantly. Plot up to 4 functions at once, zoom, pan, trace values, and find intercepts automatically.
Quick Examples
Value Table — f(x) at Specific Points
Supported Syntax Reference
Operators & Notation
x^3 → x cubed
2*x or 2x → multiplication
(x+1)*(x-2) → product
1/x → reciprocal
Functions & Constants
sqrt(x), abs(x)
log(x) → log base 10
ln(x) → natural log
exp(x) → e^x pi, e
What Is a Function Graph?
A function graph is a geometric picture of how a mathematical function behaves. Given a function y = f(x), every valid input x produces an output y. Plotting each pair (x, y) in the Cartesian coordinate plane and connecting those points traces out a curve — the graph of the function. Graphs turn abstract algebraic rules into visual shapes that reveal patterns, trends, and key properties at a glance.
Function graphs are fundamental across mathematics, science, engineering, and economics. An engineer reading a stress-strain curve, a data scientist plotting a regression line, a student sketching a parabola for a quiz — all are working with function graphs.
Reading a Function Graph: Key Features
Intercepts
The y-intercept is where the graph crosses the vertical axis; it equals f(0). The x-intercepts (or roots/zeros) are where the graph crosses the horizontal axis — the values of x for which f(x) = 0. Finding roots is one of the core problems in algebra, and the grapher locates them numerically by bisection.
Symmetry
A function is even if f(−x) = f(x) for every x in the domain. Its graph is a mirror image across the y-axis (e.g., y = x², y = cos x). A function is odd if f(−x) = −f(x) for every x. Its graph has 180° rotational symmetry about the origin (e.g., y = x³, y = sin x). Functions that satisfy neither condition are classified as neither.
Asymptotes
An asymptote is a line the graph approaches but never reaches. Vertical asymptotes occur where f(x) → ±∞ as x approaches a finite value (common in rational functions like 1/x at x = 0). Horizontal asymptotes describe the function's behavior as x → ±∞. This grapher detects discontinuities by checking for abnormally large jumps between consecutive sampled points, skipping those line segments to render asymptotes cleanly.
Continuity and Smoothness
A function is continuous on an interval if its graph can be drawn without lifting the pen. Polynomials and trigonometric functions are continuous everywhere; functions like sqrt(x) and log(x) have restricted domains. Absolute value functions produce sharp corners (non-differentiable points) rather than smooth turns.
How to Use This Function Grapher
Type any expression in the function input field using standard notation. Multiplication between a coefficient and x can be written as 2x or 2*x. Exponents use the caret: x^3. The graph updates in real time as you type, so you can watch the curve change as you adjust coefficients.
Use Zoom In (+) and Zoom Out (−) to magnify interesting regions. Drag the canvas to pan. Hover over the graph to read exact (x, f(x)) coordinates in the trace tooltip. The Reset View button returns to the default −10 to 10 window.
The Value Table section lets you enter a custom x range and step size to generate a table of (x, f(x)) pairs — useful for homework, numerical analysis, or exporting data.
Types of Functions and Their Shapes
| Function type | Example | Typical shape |
|---|---|---|
| Linear | 2x + 3 | Straight line |
| Quadratic | x^2 − 4 | Parabola |
| Cubic | x^3 − 3x | S-curve with local extrema |
| Trigonometric | sin(x), cos(x) | Periodic wave |
| Rational | 1/(x−2) | Hyperbola with asymptote |
| Square root | sqrt(x) | Half-parabola (x ≥ 0) |
| Absolute value | abs(x)+1 | V-shape |
| Exponential | exp(x) | Rapid growth curve |
| Logarithmic | ln(x) | Slowly increasing, x > 0 |
Applications in Science, Engineering, and Economics
- Physics: Projectile motion follows y = vt − ½gt² (a quadratic). Wave phenomena are modeled with sin and cos. Decay processes use exp(−kt).
- Engineering: Stress-strain curves, frequency response graphs, and control system transfer functions all rely on function graphing to visualize system behavior.
- Economics: Demand curves, supply curves, cost functions, and profit maximization problems are all graphed functions of one variable.
- Data Science: Activation functions (sigmoid, ReLU), loss curves, and probability density functions are plotted to understand model behavior.
- Mathematics Education: Graphing is the bridge between algebra and geometry — seeing a polynomial factored corresponds directly to seeing its roots on the graph.