Graphing Calculator

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Formula

Plot (x, f(x)) for 601 evenly spaced x; roots where f(x) = 0, refined by bisection on each sign change

f(x): the function you enter, written in terms of x (for example x^2 − 2x − 3)
x-intercept (root, zero): an x where f(x) = 0, the point where the graph meets the x-axis
y-intercept: the value f(0), the point where the graph meets the y-axis
bisection: halving an interval [a, b] where f(a) and f(b) have opposite signs until the root is pinned down

A graphing calculator turns an equation into a picture. Instead of working out one value at a time, it evaluates y = f(x) at hundreds of x values and draws the curve, so you can see where it crosses the axes, where it turns around and where two graphs meet. This online graphing calculator plots up to three functions at once and reads those key points off for you.

The coordinate plane: x-axis and y-axis

Graphs live on the coordinate plane, two number lines that cross at right angles:

The x-axis runs left to right. Values to the right of the center are positive and values to the left are negative.
The y-axis runs bottom to top. Values above the center are positive and values below are negative.
They meet at the origin, (0, 0).

A point (x, y) means “move x across, then y up”. For example, (3, −2) sits 3 units right and 2 units down. The axes split the plane into four quadrants. Points with x and y both positive are in quadrant I, the upper right, and the numbering continues counterclockwise.

A function such as y = −0.5x + 3 is a rule that gives one y for every x. Its graph is the set of all points (x, y) that satisfy the rule, here a straight line through (0, 3) and (6, 0).

How to use the graphing calculator

Enter a function in the Function 1 box, written in terms of x: x^2 - 2x - 3, 2x + 5, sqrt(x), abs(x - 1), e^x. You can paste y = … or f(x) = …; the prefix is ignored.
Add up to two more functions to compare graphs or find where they intersect.
Set the x-range. The default is −10 to 10. Zoom in by narrowing it and zoom out by widening it.
Leave y blank for an automatic fit, or enter both y values to fix the viewing window.
Choose radians or degrees if your functions use sin, cos or tan.

The result lists the x-intercepts first, then each function’s y-intercept, minimum and maximum, any intersections, and where the curve has gaps. The graph marks those points, and a table of values below it gives y at round x values.

Graphing a function by hand with a table of values

The calculator does the same thing you would do on graph paper, only with far more points. Take y = x² − 2x − 3 for x from −4 to 6:

x	−4	−3	−2	−1	0	1	2	3	4	5	6
y	21	12	5	0	−3	−4	−3	0	5	12	21

Plot each pair and join them with a smooth curve. The table already shows the key features:

x-intercepts (zeros): y = 0 at x = −1 and x = 3. Factoring agrees: x² − 2x − 3 = (x + 1)(x − 3).
y-intercept: at x = 0, y = −3, so the graph crosses the y-axis at (0, −3).
Minimum: the lowest value is −4 at x = 1. That point is the vertex of the parabola.
Maximum on this range: 21, at both ends, x = −4 and x = 6. A parabola that opens upward has no overall maximum, so it depends on the window.

To solve x² − 2x − 3 = 0 exactly, with every step shown, use the quadratic formula calculator.

Finding intersections

Two graphs intersect where they have the same y for the same x. Plot y = x² − 6x + 5 and y = x − 1 together and the calculator reports the intersections (1, 0) and (6, 5). It finds them as the zeros of the difference, x² − 7x + 6. Intersections are how you solve a system of equations graphically, or an equation like x² − 6x + 5 = x − 1.

Asymptotes, gaps and jumps

Not every function is one unbroken curve. y = 1/(x − 2) is undefined at x = 2, and its values shoot toward +∞ on one side and −∞ on the other. That vertical line is an asymptote. √x and ln x don’t exist for negative x, and tan x has an asymptote every π. The calculator breaks the curve at these places and lists them under “Gaps or asymptotes”. If the function grows without bound there, it reports that there is no minimum or maximum instead of printing a misleading number.

Common mistakes

Typing an equation instead of an expression. Enter only the right-hand side: 3x - 4, not y - 3x = -4. Rearrange first if you need to.
Forgetting brackets. 1/2x is (1/2)x. For 1 ÷ (2x), type 1/(2x).
A window that hides the action. If the graph looks like a flat line, the interesting part may sit outside your y-range. Clear the y-range, or narrow the x-range around the feature.
Wrong angle unit for trig. sin(x) over −10 to 10 in degrees looks almost straight. Use radians there, or widen the range to −360 to 360 in degrees.

The slope of a straight-line graph between two points is easiest with the slope calculator. For single values such as sin 40° or log 250, use the scientific calculator.

Glossary

Function: a rule that assigns exactly one y to each x.
Root, zero, x-intercept: an x where f(x) = 0.
Vertex: the turning point of a parabola, its minimum or maximum.
Asymptote: a line the graph approaches but never reaches.
Viewing window: the x- and y-ranges shown on the graph.

Worked examples

A parabola and its roots

y = x² − 4 opens upward with its vertex at (0, −4).

Inputs: Function 1: y = = x^2 - 4; x from = -10; x to = 10

f₁(x) = x^2 − 4, read as (x ^ 2) − 4 y-intercept: f₁(0) = −4 x-intercepts of f₁: where the sign of y changes between neighboring points, refined by bisection → x = −2, 2 Plotted 601 points per function from x = −10 to x = 10 (every 0.033333); y shown from −10 to 110 (automatic).

Answer: y = x^2 − 4: x-intercepts x = −2, 2.

Where a line meets a parabola

The line crosses the x-axis once; the two graphs intersect where 2x + 1 = x² − 2.

Inputs: Function 1: y = = 2x + 1; Function 2: y = = x^2 - 2; x from = -5; x to = 5

f₁(x) = 2x + 1, read as (2 × x) + 1 y-intercept: f₁(0) = 1 x-intercepts of f₁: where the sign of y changes between neighboring points, refined by bisection → x = −0.5 f₂(x) = x^2 − 2, read as (x ^ 2) − 2 y-intercept: f₂(0) = −2 x-intercepts of f₂: where the sign of y changes between neighboring points, refined by bisection → x = −1.414214, 1.414214 Intersections: solve 2x + 1 = x^2 − 2 by finding where the difference changes sign → x = −1, 3 Plotted 601 points per function from x = −5 to x = 5 (every 0.016667); y shown from −15 to 25 (automatic).

Answer: y = 2x + 1: x-intercepts x = −0.5.

A sine wave in degrees

sin(x) with x in degrees repeats every 360°.

Inputs: Function 1: y = = sin(x); x from = -360; x to = 360; Angle unit for trig = Degrees

f₁(x) = sin(x), read as sin(x) y-intercept: f₁(0) = 0 x-intercepts of f₁: where the sign of y changes between neighboring points, refined by bisection → x = −360, −180, 0, 180, 360 Plotted 601 points per function from x = −360 to x = 360 (every 1.2); y shown from −1.2 to 1.2 (automatic).

Answer: y = sin(x): x-intercepts x = −360, −180, 0, 180, 360.

A hyperbola with an asymptote (edge case)

y = 1/x is undefined at x = 0, so the curve is drawn in two pieces and never meets the x-axis.

Inputs: Function 1: y = = 1/x; x from = -5; x to = 5

f₁(x) = 1x, read as 1 ÷ x f₁ is undefined or jumps near x = 0, so the curve is drawn in separate pieces there. Plotted 601 points per function from x = −5 to x = 5 (every 0.016667); y shown from −12 to 12 (automatic).

Answer: y = 1x: x-intercepts none between x = −5 and x = 5.

Assumptions and limitations

Each function is sampled at 601 evenly spaced x values across the range you set (from −1,000,000 to 1,000,000), and the plot joins neighboring points.
x-intercepts are found where y changes sign between neighboring points and are refined by bisection to full double precision. Roots that only touch the axis, like the vertex of (x − a)², are found by searching for the smallest |y|. Two roots closer together than the sample spacing can be missed, so narrow the x-range to inspect them.
The minimum and maximum are the lowest and highest values on the chosen x-range (including its ends), refined between sample points. If the function runs off to infinity at an asymptote, there is no minimum or maximum and the result says so.
The curve is broken where the function is undefined (for example √x for x < 0) or jumps across a vertical asymptote, so no false vertical lines are drawn.
The y-range is chosen automatically to fit the curves unless you enter both y values. Spikes next to an asymptote are left off-screen.
Trig functions use radians by default. Switch to degrees to plot sin(x) with x in degrees.
Expressions follow the standard order of operations, with implicit multiplication (2x, 3(x + 1)) at the same level as × and ÷. Results are numerical, not symbolic.

Frequently asked questions

What is a graphing calculator?

A graphing calculator draws the graph of an equation such as y = x² − 2x − 3 on x- and y-axes, and lets you read off its zeros, intercepts, turning points and where two graphs cross. It also does everything a scientific calculator does.

How do you use a graphing calculator online?

Type the right-hand side of y = … in the Function 1 box, for example x^2 - 2x - 3, set the x-range, and the graph appears with its intercepts, minimum, maximum and a table of values. Add a second or third function to compare them and see where they intersect.

How do I find the zeros of a function on a graphing calculator?

The zeros are where the graph crosses or touches the x-axis. This calculator lists them automatically as x-intercepts. It finds each sign change in the y-values and narrows it down by bisection, so a zero like √2 shows as 1.414214.

What are the x-axis and the y-axis?

The x-axis is the horizontal number line and the y-axis is the vertical one. They cross at the origin (0, 0). Every point is written (x, y), meaning go x units across, then y units up or down.

Can you graph on a scientific calculator?

Usually not. A scientific calculator evaluates one expression at a time, while a graphing calculator evaluates a function at hundreds of x values and draws the curve. You can still make a table of values by hand with a scientific calculator and plot the points on graph paper.

Why is my graph drawn in separate pieces?

The function is undefined or jumps at some x values. y = 1/(x − 2) has a vertical asymptote at x = 2, tan x has one every π, and √x and ln x don't exist for negative x. The calculator leaves a gap there instead of joining the pieces with a misleading vertical line.

How do I graph in degrees?

Set the angle unit to Degrees and use a wide x-range, such as −360 to 360, so a full cycle of sin(x) or cos(x) fits on the screen. In radians, one cycle is 2π, about 6.28, wide.

Sources

2.1 The Rectangular Coordinate Systems and Graphs — College Algebra 2e — OpenStax, Rice University. Checked October 10, 2026.
3.1 Functions and Function Notation — College Algebra 2e — OpenStax, Rice University. Checked October 10, 2026.
3.3 Rates of Change and Behavior of Graphs — College Algebra 2e — OpenStax, Rice University. Checked October 10, 2026.
Bisection — Wolfram MathWorld. Checked October 10, 2026.