Free Online Graphing Calculator

How to use the graphing calculator

This free online graphing calculator is for students who need a plot without buying a handheld, and for engineers who want a quick picture of an equation. It also opens as the Graph tab beside the scientific calculator.

1. Type in the expression list above. Each row is one item and the graph redraws as you type.
2. Click a curve. Grey dots mark its roots, maxima, minima, y-intercept and crossings with other curves. Hover for coordinates.
3. Drag to pan, scroll or pinch to zoom. Share copies a link to your graph, and the download button saves a PNG.

The math behind the graphs

• Explicit, y = f(x): one y per x.
• Implicit, any equation in x and y: everything moves to one side, F(x, y) = 0, and the curve is drawn where F changes sign.
• Inequality: the same F, shaded where F < 0 or F > 0. A strict sign gives a dashed boundary, since boundary points are not in the solution.
• Polar: a point at distance r and angle θ sits at (r cos θ, r sin θ). θ runs 0 to 2π unless you add a range.
• Parametric: x and y both depend on t, also 0 to 2π by default.
• Slope field: for dy/dx = f(x, y), a short segment of slope f(x, y) is drawn at each grid point.

Roots and crossings are found by scanning the visible window for sign changes, so they only appear for what is on screen.

Example 1: where a line meets a circle

Where does y = x + 1 cross x² + y² = 25? Substitute: x² + (x + 1)² = 25, so 2x² + 2x − 24 = 0, with discriminant (b² − 4ac) 196 and roots x = 3 and x = −4. The points are (3, 4) and (−4, −3), and both satisfy the circle: 9 + 16 = 25 and 16 + 9 = 25. The chord between them is 7√2 ≈ 9.899495 long.

Crossing dots appear for curves written y = f(x), so draw the circle as two halves, y = sqrt(25 - x^2) and y = -sqrt(25 - x^2), and make the line y = x + a with a slider a = 1. Click the line to see both points. Then drag the slider: the discriminant becomes 200 − 4a².

The line is tangent (one touching point) when the discriminant is 0, at a = 5√2 ≈ 7.071068.

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Example 2: a polar rose, drawn once

Type r = cos(3θ). On the default 0 to 2π you get three petals, traced twice: the point at θ + π equals the point at θ, because cos(3θ + 3π) = −cos(3θ) and the negative r flips the point across the origin. Computed over a full sweep, the gap between the two is about 4 × 10⁻¹⁵, which is zero. One pass of length π is enough:

r = cos(3θ) {0 < θ < π}

The petal tips are at θ = 0 (r = 1, point (1, 0)), θ = 2π/3 (point (−0.5, 0.866025)) and θ = π/3, where r = −1 puts the tip at (−0.5, −0.866025). The curve returns to the origin at θ = π/6, π/2 and 5π/6.

The range changes area. Area is ½∫r² dθ: over 0 to π it is π/4 ≈ 0.7853982, the true area, and over 0 to 2π it is 1.570796, double, because the rose is counted twice. For r = cos(kθ), odd k repeats after π, but even k does not: r = cos(2θ) has four petals and needs the full 2π.

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Example 3: shading a feasible region

Constraints: x ≥ 0, y ≥ 0, x + y ≤ 6, 2x + y ≤ 8. Type each as a row (use >= and <=). Each row shades its own half-plane, and the feasible region, where all four hold, is the darkest overlap.

Corners are where two boundaries meet. Subtracting x + y = 6 from 2x + y = 8 gives x = 2, y = 4. The others are (0, 0), (4, 0) and (0, 6). The grapher does not list corners, so add the points row (0, 0), (4, 0), (2, 4), (0, 6) and check each dot lands on a corner.

To maximise 4x + 3y, test the corners: 0, 16, 20 and 18. The best is 20, at (2, 4). The area of the region is 14. A spot check: (1, 1) is feasible, but (4, 3) is not, since 4 + 3 = 7 exceeds 6.

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Example 4: a slope field with a solution curve

For dy/dx = x − y the slope at (0, 1) is −1, at (2, 0) it is 2, and on y = x it is 0. Solving by hand with y(0) = 1 gives y = x − 1 + 2e^(−x). Check: y′ = 1 − 2e^(−x), and x − y is the same. The grapher draws the field but does not solve it, so overlay your solution and watch it follow the segments. Values: y(1) = 2/e ≈ 0.7357589, y(2) ≈ 1.270671. Click the curve and its minimum sits at x = ln 2 ≈ 0.693147, on y = x, where the field is flat.

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Reading the result and common mistakes

• Radians versus degrees. sin(30) is −0.9880316 in RAD and 0.5 in DEG. In DEG, polar angles and the parametric t run from 0 to 360 instead of 0 to 2π, so r = cos(3θ) still draws its rose; write any range in degrees too, like {0 < θ < 180}. See degrees vs radians on a calculator.
• Missing grey dots. Roots, extrema and crossings are found for curves written y = f(x). An implicit circle or polar curve gets none, so hover for coordinates or split it, as in example 1.
• Off-screen roots. Zoom out before deciding a function has none.
• Dashed or solid. A dashed boundary excludes its points (<, >); solid includes them.
• log versus ln. log is base 10, ln is natural, and log_2(x) is base 2.
• "Unknown name". Add a slider row such as a = 1.

Frequently asked questions

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