Graphing Calculator for Calculus

This graphing calculator online is built for calculus rather than general algebra. Type a function and it plots ( f(x) ) together with its derivative ( f'(x) ), a tangent line you can slide along the curve, and the shaded signed area between the curve and the x-axis on an interval ([a, b]). Turn on Riemann rectangles to watch left, right, midpoint or trapezoid sums approach the exact integral.

It’s a quick function grapher for homework, lecture demos or building intuition before an exam. It runs in your browser, works on phones, and is free with no sign-up.

How to use the graphing calculator online

1. Type your function in the f(x) = box. Use ^ for powers, sqrt(), sin(), cos(), ln(), e^(x) and pi. Implicit multiplication works, so 3x and x sin(1/x) are fine. The graph redraws as you type, or tap one of the example chips.
2. Use the Layers checkboxes to show or hide the derivative (dashed), the tangent line, the area under the curve, and Riemann rectangles.
3. Drag the Tangent point x₀ slider to move the tangent line. Also, the readout shows ( f(x_0) ), ( f'(x_0) ) and the tangent line’s equation.
4. Enter Area from a and to b as numbers. The readout gives ( int_a^b f,dx ). For Riemann sums, set Rectangles n (1 to 100) and pick a Sampling rule; the readout adds the sum and its error.
5. Navigate the view: drag to pan, press Ctrl or Cmd while scrolling to zoom, or use the +, −, Fit and Reset buttons. Hover over the graph to read ( x ) and ( f(x) ) at the cursor.

How it works

The lab differentiates your function symbolically and plots ( f'(x) ) as a second curve. The tangent line at ( x_0 ) is

$$y = f(x_0) + f'(x_0)(x - x_0)$$

The area readout is the definite integral, computed numerically. Also, it’s signed: regions above the x-axis are shaded blue and count as positive, regions below are shaded red and count as negative. The Riemann sum with ( n ) subintervals of width ( Delta x = frac{b – a}{n} ) is

$$sum_{i=1}^{n} f(x_i^*),Delta x$$

where ( x_i^* ) is the left endpoint, right endpoint or midpoint of each strip. The trapezoid option joins the endpoints with slanted tops instead.

Worked example

Start with the prefilled ( f(x) = sin x + frac{x}{2} ) on ([0, 3]).

• Derivative: ( f'(x) = cos x + frac12 ).
• Tangent at ( x_0 = 1 ): ( f(1) approx 1.3415 ) and ( f'(1) approx 1.0403 ), so the tangent line is ( y approx 1.0403x + 0.3012 ). See how to find the equation of a tangent line for the hand method, or check it with the tangent line calculator.
• Area: by the Fundamental Theorem of Calculus, ( int_0^3 f,dx = 1 – cos 3 + frac94 approx 4.2400 ).

Now try the x^3 - 3x chip. Drag ( x_0 ) to ( -1 ) and ( 1 ): the tangent goes flat, because ( f'(x) = 3x^2 – 3 = 0 ) there. Those are the local maximum ((-1, 2)) and minimum ((1, -2)), the critical points. Set ( a = -2 ), ( b = 2 ) and the blue and red regions cancel exactly, so the integral is 0. On ([0, 2]) it’s ( -2 ), because more of the area lies below the axis.

Visualizing Riemann sums

Enter x^2, set ( a = 0 ), ( b = 2 ), turn on Riemann rectangles, choose Midpoints and set ( n = 4 ). The four strips have width ( 0.5 ) and midpoints ( 0.25, 0.75, 1.25, 1.75 ), so the sum is

$$0.5left(0.25^2 + 0.75^2 + 1.25^2 + 1.75^2right) = 2.625$$

The exact area is ( frac83 approx 2.6667 ), an error of about ( 0.0417 ). Switch to left endpoints and the sum drops below the true area, since ( x^2 ) is increasing on this interval; right endpoints overshoot. Slide ( n ) up toward 100 and watch every rule converge. The guide to Riemann sums explains why the midpoint rule usually wins, and the Riemann sum calculator adds Simpson’s rule and larger values of ( n ).

Tips

• Vertical asymptotes. For 1/x the curve is drawn in two pieces. Also, an interval containing 0 shows “diverges” in the readout.
• Lost the curve? Press Fit to rescale the y-axis to your function, or Reset to return to the default window.
• Keyboard control. Click the graph, then use the arrow keys to pan and + or - to zoom.

Worked example: where two graphs meet

Plot \( y = x^2 \) and \( y = x + 2 \). They cross where \( x^2 = x + 2 \), at \( x = -1 \) and \( x = 2 \), giving the points \( (-1, 1) \) and \( (2, 4) \). Reading intersections from a graph and then confirming them algebraically is a reliable way to solve equations and to set up an area between curves problem.

What to look for on any graph

• Domain: where the function exists. \( y = \sqrt{x} \) starts at 0; \( y = \ln x \) needs \( x > 0 \).
• Intercepts: where the graph crosses the axes.
• Asymptotes: \( y = \frac{1}{x} \) approaches the axes but never touches them.
• Turning points and concavity: where the graph peaks, dips and changes its bend. See the critical points calculator.
• Symmetry: even functions mirror in the y-axis; odd functions are symmetric about the origin.

Graphing families of functions

Changing one number at a time shows what it does: in \( y = a(x - h)^2 + k \), \( a \) stretches or flips the parabola, \( h \) shifts it left or right, and \( k \) shifts it up or down. Also, the same shift rules apply to every function.

Practice problems

1. Where do \( y = 2x \) and \( y = x^2 - 3 \) meet?
2. What are the asymptotes of \( y = \frac{1}{x - 2} + 1 \)?

Answers: 1. \( x^2 - 2x - 3 = 0 \): at \( x = 3 \) and \( x = -1 \), the points \( (3, 6) \) and \( (-1, -2) \). 2. \( x = 2 \) and \( y = 1 \).

Related guides

• How to Find the Equation of a Tangent Line
• Riemann Sums: Left, Right, Midpoint and Simpson’s Rule
• Fundamental Theorem of Calculus
• Area Between Two Curves

Further reading

• The Shape of a Graph, Part I (Paul’s Online Math Notes) — what the sign of ( f’ ) says about the graph you are looking at.
• Area Problem (Paul’s Online Math Notes) — how left, right and midpoint rectangles approximate the area under a curve.

For the ideas behind reading graphs, see domain and range, polynomials and exponential functions.

FAQ

Can this graphing calculator plot the derivative?

Yes. The derivative is computed symbolically and drawn as a dashed curve, and its formula appears in the readout. Also, uncheck Derivative f′(x) to hide it.

Can I graph more than one function at once?

No. The lab plots one ( f(x) ) at a time, along with its derivative and tangent line. To compare two curves, however, switch between them in the input box.

How do I find the area under a curve?

First, check Area under curve, enter ( a ) and ( b ), and read the integral in the readout. Remember that it’s signed area; for total area, split the interval where the curve crosses the x-axis.

Is it free to use?

Yes. Also, it’s free, needs no account, and works in any modern browser on desktop or mobile.