SolveMathAI

Graphing calculator

Type a function of x. Drag to pan, scroll to zoom.

Enter to solve · Shift+Enter for a new line

  • Free
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  • Every step shown
  • Arithmetic to calculus

Type a function of x and it draws immediately. Drag the plot to move it, scroll or pinch to zoom, and use the buttons to step in and out or reset. Three functions can be on the axes at once, which is the fastest way to see where two curves meet. Write powers as x^2, roots as sqrt(x), and logs as ln(x) or log(x). The solver below finds the exact roots, vertex and intersections once you can see roughly where they are.

What to read off a graph

A picture answers four questions quickly, and each one has an exact method underneath it.

  1. Roots, where it crosses y = 0

    The x-intercepts are the solutions of f(x) = 0. A parabola crossing twice has two real roots; one that only touches has a repeated root; one that misses has complex roots.

  2. The turning point

    The highest or lowest point of a curve is where its derivative is zero. For y = ax² + bx + c it is always at x = −b/2a.

  3. Where two curves meet

    The intersection is where f(x) = g(x). Reading it off the graph gives you roughly where; solving that equation gives you exactly where.

  4. Asymptotes

    A vertical asymptote is where the denominator is zero, and the curve runs away rather than crossing. This grapher lifts the pen there rather than drawing a false vertical line through it.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

Find the roots of y = x² − 4

x2−4=0x^2 - 4 = 0
  1. Set y to zero
    x2=4x^2 = 4
  2. Take both square roots
    x=±2x = \pm 2
Answer
x=−2, 2x = -2,\ 2

On the graph these are the two points where the curve crosses the horizontal axis.

Find the vertex of y = x² − 6x + 5

y=x2−6x+5y = x^2 - 6x + 5
  1. The vertex sits at x = −b/2a
    x=62=3x = \frac{6}{2} = 3
  2. Substitute back for y
    y=9−18+5=−4y = 9 - 18 + 5 = -4
Answer
(3, −4)(3,\ -4)

Where do y = x² and y = 2x intersect?

x2=2xx^2 = 2x
  1. Set them equal and collect
    x2−2x=0x^2 - 2x = 0
  2. Factor
    x(x−2)=0x(x - 2) = 0
  3. Read both solutions
    x=0, x=2x = 0,\ x = 2
Answer
(0, 0) and (2, 4)(0,\ 0)\ \text{and}\ (2,\ 4)

Never divide both sides by x here: that throws away the solution x = 0.

What are the asymptotes of y = 1/x?

y=1xy = \frac{1}{x}
  1. The denominator is zero at
    x=0x = 0
  2. As x grows, y approaches
    00
Answer
x=0 and y=0x = 0\ \text{and}\ y = 0

Where marks get lost

Trusting a reading off the picture

A graph shows you roughly where; algebra shows you exactly where. A root that looks like 1.5 is often 3/2, and sometimes 1.4983.

Judging a curve from one window

Zoomed in, a cubic can look like a line, and a wide window can hide a turning point entirely. Zoom out once before deciding what shape it is.

Writing the function ambiguously

1/x+2 is read as (1/x) + 2, not 1/(x+2). A denominator with more than one term needs brackets.

Questions

How do I graph a function?

Type it as a function of x in the first box, for example x^2 - 4, and it draws as you type. Add a second or third to see them on the same axes.

How do I zoom and move the graph?

Drag the plot to move it, scroll or pinch to zoom, or use the plus and minus buttons. Reset puts the window back to −10 to 10 on both axes.

How do I write powers and roots?

Use ^ for a power, so x squared is x^2, and sqrt( ) for a square root. A fractional power works too: x^(1/3) is the cube root.

Can I graph trigonometric functions?

Yes. sin(x), cos(x) and tan(x) all plot, in radians, which is the convention for graphing. Zoom out on the x axis to see more than one period.

How do I find where two graphs intersect?

Put both in, and the crossing points are visible immediately. For the exact coordinates, set the two expressions equal and solve — the solver below does that step.

Why does my graph have a gap?

Because the function is undefined there. 1/x has no value at x = 0, and the pen lifts rather than drawing a vertical line through the asymptote, which would be a line that is not part of the curve.

How many functions can I graph at once?

Three, each in its own colour. That is enough to compare a function with its transformation, or to find an intersection, without the plot becoming unreadable.

Is anything I type sent anywhere?

No. The expression is parsed and plotted entirely in your browser, so it works with the network off and nothing you enter leaves the page.