Plot as many as four functions of x on the same Cartesian coordinate system. Choose the visible x and y ranges, compare color-coded curves, and inspect a table of representative values without installing graphing software.
How to graph a function
- Enter one expression in terms of
xon each line. Use^for powers and parentheses whenever the intended order could be ambiguous. - Set the x and y limits. A useful first window for many textbook examples is −10 to 10 on both axes.
- Calculate, then match each expression to its curve using the legend color.
- Inspect the value table to connect points on the curve with numerical x and y pairs.
- Tighten the window around an intercept, turning point, or intersection when you need more detail.
The Slope Calculator finds an exact line from points or slope data, while the Quadratic Formula Calculator solves the zeros of a quadratic. A graph is an excellent check on those results, but the drawing itself is not an algebraic proof.
Supported expression patterns
Use arithmetic operators +, −, *, /, and ^; constants such as pi and
e; and functions including sin, cos, tan, sqrt, abs, log, ln,
and exp. Explicit multiplication such as 2*x is the clearest input. Keep
function arguments in parentheses: sqrt(x + 4) rather than sqrt x + 4.
Reading important graph features
- An x-intercept occurs where
f(x) = 0; a y-intercept isf(0). - An intersection is a point where two plotted functions have the same value.
- A turning point is where a curve changes from increasing to decreasing or the reverse.
- A gap or break can indicate a restricted domain or discontinuity, but it can also fall between sampled points.
- A steep curve that leaves the selected y range is clipped; expand the window before concluding the function is undefined.
OpenStax's graphing guidance emphasizes using intercepts, symmetry, extrema, and end behavior together when sketching a function. The calculator supplies a numerical plot; those checks help you decide whether the picture makes sense.
Examples
Compare a wave and parabola
Enter sin(x) on the first line and x^2 / 5 on the second. Their colors in
the legend match their curves and their columns in the value table.
See a vertical discontinuity
Plot 1 / (x - 2). The expression is undefined at x = 2, so the curve splits
into two branches. Narrow the x window around 2 to inspect the behavior.
Formula
Enter expressions using familiar notation:
sin(x)
x^2 / 5
sqrt(x + 4)
1 / (x - 2)
The plot samples each function throughout the x window. Use a narrower window when you need more visual detail around an intercept or turning point.
Numerical limitations
The graph is built from sampled x-values, so it may miss a very narrow spike, closely spaced roots, or a discontinuity between samples. Trigonometric functions are evaluated in radians. Confirm precise roots or extrema with algebra or a dedicated solver, and avoid reading more decimal precision from the canvas than the value table supports.
Sources
- OpenStax Contemporary Mathematics: Graphing Functions
- OpenStax Precalculus 2e: Graphs of Linear Functions
Sampling and viewing limits
A plotted curve is a finite sample inside the selected window. Narrow spikes, holes, asymptotes, discontinuities, rapid oscillation, and behavior outside the window can be missed or visually connected. Change the window and sampling density, and confirm intercepts or extrema algebraically when they matter.
Check rates of change and accumulated area
Use the Derivative Calculator to examine a function’s symbolic rate of change, or the Integral Calculator for an antiderivative or definite integral. Plotting the function can help interpret those results, but sampled points can miss discontinuities or narrow features.