Graphing Calculator

A graphing calculator draws y = f(x) on a coordinate grid and reads the important points off the curve. With the default functions above, x^2 - 2x - 3 and x + 1, it reports a y-intercept of (0, −3), x-intercepts at x = −1 and x = 3, a minimum at (1, −4), and two points where the parabola meets the line: (−1, 0) and (4, 5). Everything runs in your browser, so the graph and the key points update as you type.

Several sister tools give exact answers to one narrower question. The Quadratic Formula Calculator returns the roots of a quadratic as exact radicals, the Vertex Calculator rewrites a parabola in vertex form, and the Derivative Calculator differentiates a function symbolically. Use this page when you want to see the whole function, compare up to four curves, or find intercepts and intersections of functions that have no tidy formula: it is a function plotter and a function calculator in one.

How to use this graphing calculator

Type a function of x into the f1 box, for example 2x^2 - 3x + 1 or sin(x) + 0.5. Multiply with * or by writing terms side by side (2x, 3sin(x), x(x - 2)), use ^ for powers and abs(x) or |x| for absolute value. A leading "y =" is ignored, and fractional powers follow the handheld convention: x^(1/3) is the real cube root, so it is drawn for negative x as well. Numbers are written without thousands separators (1000, not 1,000). Add up to three more functions; each gets its own color, and the Show box hides a curve without deleting it. If an entry cannot be read, a message under the box says what to fix.

Below the graph, type new limits into x min, x max, y min and y max (-2pi or pi/2 work too), or use Zoom in and Zoom out, which halve or double the window around its center. Fit y to graph matches the y-range to your curves, Reset returns to −10 to 10 on both axes, and the presets switch to a trig window, a close-up, the first quadrant or a wide view. Hover over the graph, or tap it, to trace every function at the same x. The blue card lists the key points and lets you choose which two functions to intersect.

How to read a graph

Every point on a curve is a pair (x, f(x)): go across to an x-value, then up or down to the curve, and the height you reach is the function's value. The axis labels give the scale (in the standard window, 2 units per labeled line on a desktop and 5 on a phone). Six features describe almost any graph, the same ones the Common Core high school standards ask students to interpret and show (HSF-IF.B.4 and HSF-IF.C.7):

• Intercepts. The y-intercept is where the graph crosses the vertical axis, at x = 0; the x-intercepts, or zeros, are where it crosses or touches the horizontal axis.
• Increasing and decreasing intervals. Reading left to right, the graph rises where the function increases and falls where it decreases.
• Turning points. A local maximum is a peak higher than every nearby point; a local minimum is a valley lower than every nearby point.
• Positive and negative intervals. Above the x-axis the function is positive, below it negative, with the zeros as boundaries.
• End behavior and asymptotes. What the graph does far left and right, and the lines it approaches without reaching, such as x = 2 and y = 1 for (x + 1)/(x − 2).
• Intersections. Their x-coordinates are exactly the solutions of f(x) = g(x), which is why graphing is a standard way to solve equations approximately (HSA-REI.D.11).

The formulas behind the key points

Each key point is the solution of a simple equation, and the calculator solves it numerically on a fixed grid across the window:

Wherever two neighboring grid values have opposite signs, a continuous function must cross zero between them, and bisection halves that interval about 50 times, to the limit of double-precision arithmetic. A sign change caused by a vertical asymptote is rejected, because the values grow instead of shrinking as the interval narrows. Peaks and valleys are spotted in the same grid and refined by golden-section search, and a valley that sits on the axis, as x^2 does at x = 0, also counts as a zero.

Worked example 1: a parabola and a line

Enter f1(x) = x^2 - 2x - 3 and f2(x) = x + 1 in the standard window. For f1 the calculator reports a y-intercept of (0, −3), x-intercepts at x = −1 and x = 3, and a local minimum at (1, −4). For f2 it reports a y-intercept of (0, 1) and a single x-intercept at x = −1. The two graphs intersect at (−1, 0) and (4, 5).

Every one of those numbers can be checked by hand. Setting x = 0 gives f1(0) = −3. Factoring gives x² − 2x − 3 = (x − 3)(x + 1), so the zeros are 3 and −1. A parabola's vertex sits at x = −b ÷ (2a) = 2 ÷ 2 = 1, and f1(1) = 1 − 2 − 3 = −4. For the intersections, set the two functions equal:

The point (−1, 0) is both an x-intercept of each function and an intersection, because the line and the parabola cross the x-axis at the same spot. The table of values (start −5, step 1) agrees: both columns read 0 at x = −1 and 5 at x = 4, and the parabola's column bottoms out at −4 when x = 1.

Worked example 2: sine and cosine in the trig window

Press the Sine and cosine example. It switches to the trig window, x from −2π to 2π and y from −4 to 4, and labels the x-axis in multiples of π. For sin(x) the calculator lists five zeros, x = −6.2832, −3.1416, 0, 3.1416 and 6.2832 (that is −2π, −π, 0, π and 2π), maxima at (−4.7124, 1) and (1.5708, 1), and minima at (−1.5708, −1) and (4.7124, −1). For cos(x) it finds zeros at ±1.5708 and ±4.7124, a maximum at (0, 1) and minima at (±3.1416, −1).

The two curves meet four times: (−5.4978, 0.7071), (−2.3562, −0.7071), (0.7854, 0.7071) and (3.927, −0.7071). Those x-values are −7π/4, −3π/4, π/4 and 5π/4, the angles where tan(x) = 1, and 0.7071 is √2 ÷ 2 rounded to four places. The example also sets the table to start at 0 with a step of pi/6, so its rows reproduce the unit-circle values: sin(π/6) = 0.5 and cos(π/6) = 0.866 in the second row, and sin(π/2) = 1 and cos(π/2) = 0 in the fourth.

Worked example 3: a rational function and its asymptotes

Rational functions show why a plotter must know when not to connect the dots. Press the Rational example: f1(x) = (x + 1)/(x − 2) and f2(x) = 1. The calculator reports a y-intercept of (0, −0.5), one x-intercept at x = −1, no turning points and no intersection, and it draws two branches with a gap at x = 2, where the denominator is zero.

The table (start 1.5, step 0.1) shows the gap in numbers: f(1.9) = −29, undefined at x = 2, then f(2.1) = 31. Closer in, f(1.99) = −299 and f(2.01) = 301. A simple plotter would join −29 and 31 with a false vertical line; this one tests each large jump by bisection and lifts the pen. The line y = 1 is the horizontal asymptote: f(100) = 1.0306 and f(−100) = 0.9706, but (x + 1)/(x − 2) = 1 leads to 1 = −2, so the graphs never meet.

Choosing a window for common function families

A graph is only as useful as its window. The standard window from −10 to 10 suits many textbook problems, but each family of functions has features that can fall outside it. The table compares the standard window with a better choice for one example per family. Every result in it was produced by this calculator.

Rules of thumb: for a line, include the y-intercept b and the x-intercept −b/m; for a parabola, center x on the vertex at −b/(2a); for growth problems, start x at 0; for trig, show one or two periods (2π divided by the coefficient of x); for rational functions, include every zero of the denominator; for logarithms, start just left of the domain boundary. Adding y = 1000 to the exponential row gives the doubling time of 6% growth, x = 11.8957, close to the rule-of-72 estimate of 12.

Key features of the parent functions

Parent functions are the simplest member of each family; most textbook functions are shifted, stretched or reflected versions of them. The first table shows what this calculator reports for ten parent functions in the standard window, the second their values at seven x-values ("undefined" means outside the domain).

Two patterns stand out. Of the functions through the origin, only x^2 and abs(x) turn there; x^3 and cbrt(x) pass straight through, so no turning point is listed. And the values at x = 4 rank the families by growth: the cube has reached 64, 2^x and x^2 are tied at 16 (they cross there), and the square root, cube root and logarithm are at 2 or below.

How this differs from a handheld graphing calculator

On a TI-84 Plus CE you type functions into the Y= editor, pick a zoom such as ZStandard (x and y from −10 to 10) or ZTrig, and use the CALC menu for key points. Finding a zero means choosing zero and answering three prompts, Left Bound?, Right Bound? and Guess?; intersect asks First curve?, Second curve? and Guess?, and Texas Instruments notes that the intersection has to be inside the window. Each run returns one point.

This page lists every zero, turning point and intersection in the window at once, with no bounds or guesses, and recomputes them whenever you edit a function or the window. Fit y to graph works like the handheld ZoomFit, which recalculates Ymin and Ymax from the lowest and highest y-values between Xmin and Xmax, except that it ignores the spikes next to vertical asymptotes. Asymptotes are always handled here; the TI-84 Plus CE has a Detect Asymptotes switch in its graph format screen.

A handheld still wins elsewhere: it is allowed in testing rooms where phones and laptops are not, needs no browser, and adds statistics plots, regressions and parametric, polar and sequence graphing. Both share one quirk: TI's own tutorial notes that ZStandard spaces the tick marks differently on the two axes, and the same is true on most screens here.

Who this graphing calculator is for

Algebra, precalculus and calculus students use it to check homework: graph a function, read off the intercepts and turning points, and compare them with the hand calculation, the skill the Common Core standards describe for graphing by hand in simple cases and with technology in harder ones. Teachers use it to project clean graphs with labeled axes, and parents and tutors use the table of values to walk through a problem one x-value at a time. Students without a handheld at home can practice the same window and trace habits they use in class.

Outside a math course, anyone facing an equation with no tidy algebraic solution, such as e^x = x + 2, can graph both sides and read the intersections: x = −1.8414 and x = 1.1462.

Key terms

• Function: a rule that gives exactly one output f(x) for each input x; its graph passes the vertical line test.
• Window: the visible rectangle, set by x min, x max, y min and y max.
• Zero, root or x-intercept: an x-value where f(x) = 0.
• Local maximum and minimum: peaks and valleys compared with nearby points (turning points or relative extrema), not necessarily the highest or lowest overall.
• Asymptote: a line the graph approaches without reaching, such as x = 2 for (x + 1)/(x − 2).
• Domain: the x-values where a function is defined; sqrt(x) needs x ≥ 0 and ln(x) needs x > 0.
• Parent function: the simplest function in a family, such as x^2 for quadratics.

What changes the result

• The window. Key points are searched only for x between x min and x max. Points above or below the visible y-range are still listed, but not drawn.
• How you write the function. 1/2x means (1/2)·x and -x^2 means −(x^2). When in doubt, add parentheses.
• Radians. sin(30) is the sine of 30 radians, −0.988, not of 30 degrees.
• The domain. sqrt and ln return no value for negative inputs, so their graphs simply stop.

Tips: when a curve disappears, press Reset and then Fit y to graph; zoom in on a cluster of zeros to separate them; and hide a function with its Show box instead of deleting it when you compare intersections pair by pair.

Limitations

Every entry must be y as a function of x, so circles have to be split into two halves and parametric, polar and inequality graphs are not supported. Results are numerical: the calculator reports 2.4495 rather than √6, so use the exact-answer calculators below when you need radicals.

With a 2,001-point grid, two zeros closer together than 1/2,000 of the window width can be missed, and fast oscillations such as sin(100x) in a wide window are undersampled; zooming in fixes both. At most 60 zeros and 120 turning points are refined per function. Endpoints of the window or of a domain, such as the start of sqrt(x) at the origin, are not counted as turning points.

Related calculators

• To turn two points into a slope and the equation of a line, or to check the slope of a linear graph, use the Slope Calculator.
• For the exact roots of ax² + bx + c = 0, including complex roots that never show up as x-intercepts, use the Quadratic Formula Calculator.
• For a parabola's exact vertex, axis of symmetry and vertex form, use the Vertex Calculator.
• For exact x- and y-intercepts of a line or a quadratic with the algebra written out, use the X and Y Intercept Calculator.
• To locate turning points exactly, differentiate with the Derivative Calculator: a smooth function turns where its derivative equals zero.
• For sine, cosine and tangent of a single angle in degrees or radians, use the Trigonometry Calculator.

Sources

Graphs and key points are computed from your own functions with standard numerical methods, so no external data is involved. These primary sources document the conventions and the handheld comparison:

• NIST Digital Library of Mathematical Functions, §4.2 Logarithms (log_a z = ln z / ln a, eq. 4.2.8, and log₁₀ z = ln z / ln 10, eq. 4.2.15) and §4.14 Trigonometric functions (tan = sin / cos, periods 2π and π).
• Common Core State Standards for Mathematics, Interpreting Functions, HSF-IF.B.4 and HSF-IF.C.7, and Reasoning with Equations and Inequalities, HSA-REI.D.11.
• Texas Instruments, Solution 34516: Finding the Zero of a Function on the TI-84 Plus CE and Solution 34471: Finding the Intersection of Two Functions.
• Texas Instruments, TI-84 Plus CE reference: ZOOM menu commands (ZStandard, ZTrig, ZoomFit, ZSquare), the "Graphing an Equation" tutorial (standard window from −10 to 10), and the Getting Started with the TI-84 Plus CE guidebook (graph format screen with Detect Asymptotes).
• College Board, SAT Suite calculator policy (embedded graphing calculator, approved and prohibited devices).