Function graph: zeros, extrema and asymptotes
The grapher draws a function from its formula and immediately answers the questions graphs are drawn for: where the function is zero, where its turning points and asymptotes are, where two graphs cross and on which intervals it increases. All calculations run in the browser, so the graph redraws on every keystroke and every slider move.
How to graph a function
Type a formula such as x^2 - 4 or y = kx + b. Enter adds a new row — another function on the same axes.
Read the answers under the formulas: tapping a number shows the point. Drag the graph to move it; zoom with the wheel or two fingers.
Open “Analyze the function” for intervals, the tangent and the area, then save the result as PNG, CSV or a link.
Type a formula to see its zeros, turning points and intersections
Drag to move · wheel or pinch to zoom · tap for a point
The PDF analysis sheet — graph, answers, intervals and a table of values on one printable page — is part of Premium. What Premium includes
Features
What you can graph
- Parabolas and polynomials: x^2 - 4, x^3 - 3x, a(x - m)^2 + n with sliders
- Rational functions with asymptotes: 1/x, (2x + 1)/(x - 3), x + 1/x
- Trigonometry with the x axis labeled in π: sin x, tan x, 2sin(3x)
- Logarithms and exponentials: ln x, 2^x, e^(-x)
- Inequalities and circles: y > x^2 - 2, x^2 + y^2 = 9
- Polar and parametric curves: r = 2sin(3θ), (cos 3t, sin 2t)
What the input field understands
Numbers with a point or a comma, implicit multiplication (2x, kx, 3sin(x), (x+1)(x-1)), functions without brackets (sinx, ln x), powers with ^ and superscripts ², ³, absolute value with vertical bars. Functions: sin, cos, tan, cot, asin, acos, atan, sinh, cosh, tanh, ln, log10, log2, log(base, number), sqrt, cbrt, abs, exp, sign, floor, ceil, round, min, max. Constants pi and e, angles in radians. The CSV table uses a decimal point and commas between columns, which is what English-language spreadsheets expect.
How this differs from a plain grapher
Most graphers draw the curve and stop there: where it crosses the axis and where its vertex is has to be read off the grid by eye. Here those numbers are computed and listed next to the graph, together with asymptotes, intersections and monotonicity intervals — so a “sketch and analyze the function” answer can be checked without redoing everything by hand. A parameter in the formula becomes a slider, so a question like “for which k does the line touch the parabola” can be answered by moving your hand. And tangent graphs have no vertical stroke that the function does not have.
Frequently asked questions
Why is there no vertical line where tan x breaks?
Because the function has no such line. At π/2 the tangent is undefined: on the left it goes to plus infinity and on the right it comes back from minus infinity. A grapher that joins those two pieces with a segment draws a line that does not exist, and it looks like part of the graph. Here that jump is detected, the pen is lifted, and the asymptote x = π/2 is drawn as a thin dashed line and listed in the answers.
How do I get a slider for a parameter such as k in y = kx + b?
Just type the letter in the formula. Any single Latin letter other than x, y and e becomes a parameter: a slider with its value and editable limits appears under the formulas. The ▶ button sweeps the parameter from its minimum to its maximum and back so you can watch the graph change. Parameter values are stored in the link, so someone else opens the graph with the same k and b.
How do I draw a circle, an inequality or a polar curve?
The way a textbook writes them. x^2 + y^2 = 9 is a circle of radius 3 (implicit curves use equal scales on both axes so a circle does not look like an ellipse); y > x^2 - 2 is the region above the parabola, shaded in the function color, with a dashed boundary for a strict inequality. r = 2sin(3θ) is a polar rose and (cos t, sin t) a parametric curve; the t limits can be changed under the row.
How accurate are the zeros, turning points and asymptotes?
Zeros are refined by bisection to six decimal places and checked by substitution; turning points use golden-section search. A vertical asymptote is told apart from a tall peak by growth: near an asymptote the value grows without bound, near a peak it does not. All of this is searched in the visible range, while symmetry, period and asymptotes at infinity are checked by substituting many points. These are numerical answers: good for checking a solution, not a replacement for writing it out.
Can I type 2x, kx and sinx without multiplication signs or brackets?
Yes. The field reads school notation: 2x, kx + b, 3sin(x), sinx, (x+1)(x-1), |x|, x². If an expression still cannot be read, the message under the field names the unknown name and the character where reading stopped.
Why does the graph sometimes flatten into a line?
That happens with a function that has an asymptote: one huge value stretches the vertical axis and everything else is squeezed towards zero. That is why the top and bottom two percent of values are ignored when fitting the Y axis. If that is not enough, switch off the arrows button “Fit the Y axis” on the right of the graph and move or zoom the graph yourself.
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