Graphing Calculator

Q: Why does the graph have gaps?

Gaps appear where the function is undefined, such as √(x) for x < 0, or at vertical asymptotes like x = 0 for 1/x.

Q: Can I solve equations with the graph?

Yes, approximately: graph each side and read the x-coordinates where the curves cross. For exact answers, use the equation solver.

Q: Does it work on phones?

Yes. The graph resizes to the screen and supports touch dragging.

Q: How do I find where two graphs intersect?

Enter each function in its own row and zoom in on the crossing point. The x-coordinate solves the equation formed by setting the two functions equal, and the y-coordinate is their common value there.

About the Graphing Calculator

A graph turns a formula into a picture: roots become x-intercepts, maxima and minima become peaks and valleys, and asymptotes show up as places the curve races off to infinity.

Inequalities such as $y > 2x + 1$ are drawn as a shaded region: a solid boundary means points on the line are included ( $\le$ , $\ge$ ) and a dashed boundary means they are not ( $<$ , $>$ ). Points such as $(3, 0)$ are drawn as labelled dots, which is handy for checking roots and intercepts.

Enter up to five rows at once to compare them. The plotter samples each curve finely, breaks it at gaps and vertical asymptotes instead of drawing false connecting lines, and redraws instantly as you zoom or drag.

What is a graphing calculator?

A graphing calculator draws the set of points that satisfy a relation. For a function $y = f(x)$ it computes $y$ for many closely spaced values of $x$ across the visible window and joins them into a curve. For an inequality it shades every point of the plane that makes the statement true, and for a list of points it simply marks each coordinate pair.

Seeing the picture answers questions that are slow to settle by algebra alone: how many solutions an equation has, roughly where they are, whether a function is increasing or decreasing, and how it behaves for very large or very small $x$ . This plotter runs entirely in your browser, so nothing you type is sent anywhere.

How to use the Graphing Calculator

Type a function of x, such as x^2 - 2x - 3 or sin(x). You may write y = or f(x) = in front.
For an inequality, put y alone on one side, for example y >= x^2 - 4 or y < 3 - x. For a vertical boundary, type x < 2 or x >= -1.
To plot points, type them in brackets separated by commas, for example (1, 2), (-3, 0.5).
Add more rows with the Add entry button to compare them in different colors.
Drag the graph to pan, use the mouse wheel or the + and − buttons to zoom, and press Reset to return to the default view.

Formulas

Slope-intercept line

y = mx + b

Vertex form of a parabola

y = a(x - h)^2 + k

Sinusoid

y = A\sin(Bx + C) + D

Worked example

As an example, enter $y = x^2 - 2x - 3$ . The graph is an upward parabola because the coefficient of $x^2$ is positive. It crosses the x-axis at $x = -1$ and $x = 3$ , which are the roots, since $x^2 - 2x - 3 = (x + 1)(x - 3)$ . It crosses the y-axis at $-3$ , the constant term, and its lowest point, the vertex, is at $(1, -4)$ , halfway between the roots. Add a second row with $y < x^2 - 2x - 3$ and the region below the curve is shaded with a dashed boundary.

Detailed explanation

Reading key features from a graph

The x-intercepts of $y = f(x)$ are the real solutions of $f(x) = 0$ , and the y-intercept is $f(0)$ . Peaks and valleys are local maxima and minima, where the derivative is zero. A curve that rises without bound or approaches a horizontal line far to the left and right reveals the end behaviour. Vertical asymptotes appear where a rational function's denominator is zero, such as $x = 2$ for $y = \frac{1}{x - 2}$ ; the plotter breaks the curve there rather than drawing a false vertical line across the gap.

Transformations

Graphing a function next to a modified version is the quickest way to learn transformations. Adding $k$ outside, $f(x) + k$ , shifts the graph up by $k$ . Replacing $x$ with $x - h$ shifts it right by $h$ . Multiplying by $a$ stretches it vertically, and a negative $a$ also reflects it in the x-axis. Try $y = x^2$ , $y = (x - 2)^2$ and $y = -x^2 + 3$ in three rows to see all three effects at once.

Solving equations and systems graphically

To solve an equation such as $2^x = x + 3$ , graph each side as its own row and look for intersections; their x-coordinates are the solutions. The same approach solves a system of two equations in two unknowns. Graphs give approximate values, so use them to see how many solutions exist and roughly where, then confirm with the equation solver or the system of equations calculator for exact answers.

Common mistakes

Typing sin x without brackets; use sin(x).
Expecting $\sqrt{x}$ or $\ln x$ to appear for negative x; they are only defined for $x \ge 0$ and $x > 0$ .
Missing features because the window is too small; zoom out to see the overall shape first.
Reading an approximate intersection from the graph as an exact answer.

Tips

Graph both sides of an equation as separate functions; their intersections are the solutions.
Compare $f(x)$ with $f(x) + k$ or $f(x - h)$ to see translations.
Use the derivative calculator alongside to check where the graph is increasing.
Zoom in on an intercept or intersection to read its coordinates more precisely.
Plot a known point as well as the curve to check that the curve passes through it.

Frequently asked questions

What can I graph?

Any function of x built from numbers, + − × ÷, powers, roots, exponentials, logarithms, absolute value and trig, inverse trig and hyperbolic functions.

How many functions can I plot at once?

Up to five rows. Each row is a function, an inequality or a list of points, drawn in its own color.

How do I graph an inequality?

Type it with y alone on one side, such as y > 2x + 1 or y <= x^2. The solution region is shaded; a dashed boundary is not part of the solution and a solid one is. For vertical boundaries type x < 3 or x >= -2.

Can I plot points?

Yes. Type one or more points in brackets, such as (2, 3) or (0, 0), (1, 1), (2, 4). Each point is drawn as a labelled dot, up to 50 per row.

How do I zoom and pan?

Scroll to zoom around the pointer, use the + and − buttons, and drag to move the view. Reset restores the window from −10 to 10.

Why does the graph have gaps?

Gaps appear where the function is undefined, such as $\sqrt{x}$ for $x < 0$ , or at vertical asymptotes like $x = 0$ for $\frac1x$ .

Can I solve equations with the graph?