semkuGraphs of functions calculator
Plot functions and see roots and intersections.
The graph may miss rapid oscillations in this window; zoom in.
| x | f₁(x) | f₂(x) |
|---|---|---|
| −2 | 5 | −1 |
| −1 | 0 | 0 |
| 0 | −3 | 1 |
| 1 | −4 | 2 |
| 2 | −3 | 3 |
| 3 | 0 | 4 |
| 4 | 5 | 5 |
Calculation rules
Enter one to five functions in the form y = f(x), using the expression after the shown y =. Supported operators are +, −, *, / and ^, with parentheses and implicit multiplication such as 2x, 3(x+1) and 2sin(x). The minus signs - and − and multiplication signs × and · are accepted. Functions are sin, cos, tan, asin, acos, atan, sqrt, abs, ln, log and exp; constants are pi (π) and e. Use parentheses for function arguments. Trigonometric angles are in radians; log is base 10 and ln is the natural logarithm. Powers associate to the right: 2^3^2 = 512. A leading minus applies after the power: −x^2 means −(x^2).
Graphs, roots and intersections are searched within the visible window. Move or zoom the window to search elsewhere. A y-intercept is evaluated at x = 0 even when that point is outside the window, and is then marked “outside this window.” Where a function has no value, the graph has a gap and the value table shows “No value.” Points also leave gaps and show “Not computed” in the value table when a negative base’s exponent cannot be determined exactly, a value is too large to represent, or rounding makes it unclear whether an input is allowed; this does not mean the function has no value there.
Plotting and searching use floating-point arithmetic with about 15–16 significant digits. Found coordinates use ≈ by default. The = sign is reserved for exact rational verification of expressions built from number literals, x, arithmetic operators and integer powers, with no function calls or pi or e. If exact verification reaches its limits, the coordinate remains approximate. Coordinates are rounded once to the selected decimal places; halfway values are rounded away from zero. Floating-point cancellation can produce false candidates in expressions outside the rational verification path.
The search checks sign changes between samples and refines candidates, rejecting poles such as x = 0 for 1/x. Roots that touch the x-axis without crossing it may be missed; touching candidates whose validity is uncertain are labelled separately. “No roots found” does not mean no roots exist, and the same distinction applies to intersections. Excessive oscillation or a search stopped early is reported beside the result.
A zero function or coincident pair is shown as a window-wide state instead of a list of coordinates. These states are based on sampled values; rational expressions are checked exactly at those samples, while other expressions use “appears” or “appear.” Sampling does not prove what happens between samples, and a function with no value is not a zero function.
PNG and SVG downloads show the current graph window, colors and axis ticks. The share link contains the entered functions and graph window. The prefilled functions and value table are examples.
Frequently asked questions
- What are the roots of x^2 − 2x − 3?
- The roots are x = −1 and x = 3, and the y-intercept is (0, −3). With x + 1 also plotted, the intersections are (−1, 0) and (4, 5). These are the example functions.
- Why do some coordinates use ≈?
- The graph search is numerical, and coordinates are approximate unless the rational verification path proves the displayed values exactly. Expressions containing functions such as sin or sqrt stay approximate under this rule, including coordinates displayed as zero.
- Does “No roots found” mean there are no roots?
- No. The search can miss roots between samples, especially roots that touch the axis without changing sign. Move or zoom the window to search again; possible touching roots are labelled as candidates.
- Why is a y-intercept marked outside this window?
- The y-intercept is the function's value at x = 0. It is still listed when that point lies outside the current graph window.
- Can I download or share the graph?
- Download PNG or SVG for the current window, colors and axis ticks, or copy a share link containing the functions and graph window.
- Can I graph x^2 + y^2 = 1 or a list of points?
- This calculator accepts functions in y = f(x) form. Implicit equations, polar and parametric equations, inequality regions and point lists are outside its scope.
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For details, see the privacy policy in Korean.
Calculations run in your browser without sending your calculation inputs or results to the server. See “Privacy on this page” for information about access data.