Graph plotter: type an equation, see the graph

A graph plotter draws the graph of an equation so you can see its shape: where it crosses the axes, where it turns, and how steep it is. Type y as a function of x, such as x² − 4x + 3, and our graph plotter, the Custom Graph, draws it straight away. It then lists the x and y intercepts and finds the turning points, the derivative and the area under the curve between two values of x. You can zoom and pan anywhere on the curve.

Try it

Custom Graph

Plot any equation with full analysis tools

Open the calculatorGo to its page

What the graph plotter does

• Plots y = f(x): straight lines, quadratics, cubics and other polynomials, reciprocals, exponentials, logarithms, roots, trig functions and combinations of them.
• Intercepts: a table of the y-intercept and the x-intercepts (roots), marked on the graph.
• Analysis: f′(x) gives the derivative, with + and − buttons for the second, third and higher derivatives; ∫ dx gives the area between two x values; Min / Max finds the turning points.
• Practice: Random problem plots a new equation, asks a question about it and checks your answer.
• Sharing: a link that opens your exact graph.

It plots one equation of the form y = f(x) at a time. It can't draw several curves on the same axes, vertical lines such as x = 3, circles and other equations written in both x and y (like x² + y² = 25), parametric or polar curves, inequalities, or points from a table of data. For a straight line, quadratic, cubic or logarithm with sliders for each coefficient, use the Linear Graph, Quadratic Graph, Cubic Graph or Logarithmic Graph.

How to plot a graph

1. Open the Custom Graph and type the right-hand side of y = … in Equation. x^2 becomes x², and sin, cos and sqrt are recognised as you type. The question mark beside it lists the syntax.
2. Scroll or pinch to zoom, and drag to pan. The zoom buttons and a reset are under Display, with the colour theme, the background grid and a switch for the intercept markers.
3. Read the Intercepts table.
4. Under Analysis, choose f′(x), ∫ dx or Min / Max. For the area, type the two limits into the integral it shows.
5. Press Random problem under Practice and help to test yourself.

How to type each function

One thing to watch: in this plotter log is the natural logarithm, the same as ln, so type log10(x) when you want base 10.

Worked examples you can check in the graph plotter

Example 1: a quadratic

Type x^2 - 4x + 3. The graph is a U-shaped parabola. The Intercepts table shows the y-intercept (0, 3) and the roots (1, 0) and (3, 0), because x² − 4x + 3 = (x − 1)(x − 3). Min / Max gives a local minimum at f(2) = −1, the vertex, and f′(x) gives 2x − 4, which is 0 at x = 2.

On this graph x is across and y = x² − 4x + 3 is up. The roots sit at 1 and 3 on the x-axis, and the vertex is halfway between them.

Choose ∫ dx and set the limits to 1 and 3: the plotter gives −1.333328. The area between the roots is below the axis, so it comes out negative. The exact value is −4/3; the plotter adds up thin strips, so the last digits are approximate. The area under curve calculator guide shows how to get exact areas.

Example 2: a cubic with two turning points

Type 2x^3 - 9x^2 + 12x - 2. Min / Max finds a local maximum at f(1) = 3 and a local minimum at f(2) = 2. Choose f′(x) and the plotter gives 6x² − 18x + 12, which factorises to 6(x − 1)(x − 2): zero at exactly those two x values. Press + for the second derivative, 12x − 18, which is negative at x = 1 (a maximum) and positive at x = 2 (a minimum). The graph has a single root, at x = 0.193556067641, and its y-intercept is (0, −2).

Example 3: the path of a ball

A ball is kicked along the path y = −0.1x² + 2x, with x and y in metres. Type -0.1x^2 + 2x. The roots are (0, 0), where it was kicked, and (20, 0), where it lands 20 m away. Min / Max gives a local maximum at f(10) = 10: its highest point is 10 m up, halfway along.

Example 4: graphs with no intercepts

Type 1/x. The table says no roots and no y-intercept: the curve never touches either axis, because 1/x is never 0 and isn't defined at x = 0. Now type 2^x: there are still no roots, since 2ˣ is always positive, but the y-intercept is (0, 1).

Example 5: a graph with many roots

Type sin(x). It crosses the axis at every multiple of π, so it has infinitely many roots. The table lists 10 of them, from −18.849555921539 to 9.424777960769. The plotter looks for roots and turning points only between x = −20 and x = 20, and lists at most 10 roots.

Reading a graph

• Roots (x-intercepts): the solutions of f(x) = 0. Plotting is a quick way to solve an equation you can't factorise.
• y-intercept: f(0), the value when x is 0.
• Turning points: where the gradient is 0. A local maximum is higher than the points either side; a local minimum is lower.
• Gradient: the derivative f′(x). Positive means the curve is going up from left to right, negative means it is going down.
• Asymptotes: lines the curve gets closer and closer to but never meets, such as both axes for y = 1/x.

Common mistakes

• Typing "y =". Type only the right-hand side: x^2 - 4x + 3.
• Degrees in trig. sin(x) uses radians. To plot sin x°, type sin(x*pi/180).
• Missing brackets. 1/x+1 is (1/x) + 1. Type 1/(x+1) for one over x + 1.
• log for base 10. Here log is the natural log. Use log10(x).
• Expecting every root. Roots beyond x = ±20 aren't listed. Pan along the graph to see them.

Read next

Maths · GraphsLinear graph calculator: plot y = mx + b, find interceptsFree linear graph calculator: plot y = mx + b, change the slope and intercept, and read where the line crosses each axis, with practice questions.