Graphing Calculator
Plot one or two functions of x over any window, with the roots, intercepts and turning points found for you, and a table of values you can read off.
Graphing Calculator: with the default inputs, y at that x is -4.
An expression in x. Brackets, + − * / ^, and functions like sqrt, abs, sin, cos, ln, log, exp.
A second function drawn on the same axes. Leave blank for one curve.
The single point the headline answer is for.
- Second function at that x
- —
- Roots in the window
- -2, 2Where the first curve crosses the x-axis.
- y-intercept
- (0, -4)
- Lowest point in the window
- (0, -4)
- Highest point in the window
- (-10, 96)
- Where the two curves meet
- —
Assumptions
- The curve is sampled at 241 points across the window; roots and turning points are found from those samples, so features narrower than one step can be missed.
- Trigonometric functions take radians.
- Points where the function has no value are omitted and the line is drawn straight across the gap.
| x | y = x^2 - 4 |
|---|---|
| -10 | 96 |
| -9 | 77 |
| -8 | 60 |
| -7 | 45 |
| -6 | 32 |
| -5 | 21 |
| -4 | 12 |
| -3 | 5 |
| -2 | 0 |
| -1 | -3 |
| 0 | -4 |
| 1 | -3 |
| 2 | 0 |
| 3 | 5 |
| 4 | 12 |
| 5 | 21 |
| 6 | 32 |
| 7 | 45 |
| 8 | 60 |
| 9 | 77 |
| 10 | 96 |
How this is worked out
The formula
The curve is drawn by evaluating y = f(x) at 241 evenly spaced values of x across the window. Roots: wherever two neighbouring samples have opposite signs, the crossing is narrowed by bisection. Intersections: the same search run on f(x) − g(x), which is zero exactly where the two curves meet.
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- y =
- An expression in x. Brackets, + − * / ^, and functions like sqrt, abs, sin, cos, ln, log, exp.up to 200 characters · defaults to "x^2 - 4"
- and y = (optional)
- A second function drawn on the same axes. Leave blank for one curve.up to 200 characters · defaults to ""
- Window from x =
- A number.defaults to -10
- to x =
- A number.defaults to 10
- Evaluate at x =
- The single point the headline answer is for.defaults to 0
What you get back
- y at that xmain answer
- Second function at that x
- Roots in the window
- Where the first curve crosses the x-axis.
- y-intercept
- Lowest point in the window
- Highest point in the window
- Where the two curves meet
What this assumes
- The curve is sampled at 241 points across the window; roots and turning points are found from those samples, so features narrower than one step can be missed.
- Trigonometric functions take radians.
- Points where the function has no value are omitted and the line is drawn straight across the gap.
About this calculator
Type a function of x, choose the window, and read the curve — along with the numbers you would otherwise hunt for by eye: where it crosses zero, where it crosses the y-axis, its highest and lowest points in view, and where two curves meet.
Writing the function
Use x as the variable and ordinary notation: x^2 - 4, 3*x + 1, sqrt(x), sin(x), ln(x), abs(x - 2), (x^2 - 1)/(x - 1). Multiplication needs its star — write 3*x, not 3x. Angles are in radians, so sin(pi/2) is 1.
What the numbers mean
- Roots are where the curve crosses the x-axis, found by sampling. A curve that only touches zero without crossing —
x^2at the origin — has a root the sign test cannot see, and two roots closer together than one sample step can be missed as a pair. Narrow the window and they appear. - Lowest and highest point are the smallest and largest sampled values inside the window, not the true turning points. They land on a turning point when one is in view and on the window edge when the curve is still climbing.
- Where the two curves meet solves f(x) − g(x) = 0, which is the same problem as finding a root.
Gaps and asymptotes
Some functions have no value at some x: 1/x at zero, sqrt(x) below zero, ln(x) at or below zero. Those points are left out, and the line is drawn straight across the gap rather than broken — so a vertical asymptote looks like a steep segment, not a wall. Near one, the curve also runs to enormous values; the plot clips those so the rest of the curve stays readable, and says so when it does.
What this is not
The window is set by typing numbers, not by dragging. There is no slider, no parametric or polar mode, and no implicit curve like x² + y² = 25. For those, a dedicated graphing tool is the right thing to reach for. What this does is plot an explicit y = f(x) quickly, and hand you the numbers that go with it.
Frequently asked questions
▸Why does 3x not work?
Multiplication needs to be written out: 3x. It is the one place the notation differs from what you would write on paper, and it is what lets the parser tell 3x from a variable called x3.
▸How do I graph two functions at once?
Put the second one in the optional field. Both are drawn on the same axes, and the calculator reports every x where they meet inside the window.
▸Why is a root missing?
Roots are found where the curve changes sign between two sampled points. A curve that touches zero and turns back — y = x² at the origin — never changes sign, so the test cannot see it. Two roots very close together can also fall between samples. Narrowing the window increases the sampling density and usually reveals them.
▸Why does the graph look clipped?
Near a vertical asymptote the curve runs to enormous values, which would squash everything else flat. Points far outside the bulk of the curve are left off the plot so the rest stays readable; a warning says when that has happened.
▸Are angles in degrees or radians?
Radians, which is the mathematical convention. To work in degrees, convert inside the expression: sin(x * pi / 180).
▸Can it graph a circle or a polar curve?
No. It plots explicit functions of the form y = f(x), one y for each x. A circle fails that test, because x² + y² = 25 has two y values for most x.
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The questions people ask next to a graphing.
Type a whole expression and get an exact answer: brackets, powers, roots, logs, trigonometry in degrees or radians, and the order of operations applied properly.
Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, discriminant, vertex, axis of symmetry, factored form and a graph of the parabola.
Slope, y-intercept and the equation of the line through two points, plus distance, midpoint, angle and the perpendicular slope — with rise-over-run shown.
Raise any base to any power — negative, fractional or huge — with the expanded multiplication, scientific notation and the rule that applies shown.
Logarithm of any number to base 10, e, 2 or any base you choose — with the change-of-base step, an antilog check, and ln, log₁₀ and log₂ side by side.