A polynomial graph plotter turns a list of coefficients into a curve, and then tells you the four things the curve is actually being asked about: where it crosses the x-axis, where it turns, where it crosses the y-axis, and what it does at the far left and far right. This tool finds all four numerically and marks them on the drawing rather than leaving you to read them off by eye.
Arb Digital built this page around the viewing window, which is the part most plotters get wrong. A fixed window from −10 to 10 hides everything about a polynomial whose roots sit at 40 and 60, and it flattens one whose interesting behaviour all happens between −0.2 and 0.3. This plotter finds the roots and turning points first, then chooses a window that contains all of them with a margin — so the picture you get is the picture of the polynomial, not of an arbitrary rectangle.
What This Polynomial Plotter Does
Enter the coefficients from the highest power down and the tool evaluates the polynomial with Horner's scheme, locates every real root by bracketing sign changes and refining them by bisection, differentiates the polynomial exactly and repeats the process on the derivative to find the turning points, and then frames a window around everything it found.
It reports the real roots as the headline result, and gives the degree, the y-intercept, the number of turning points and the value at any x you choose. The breakdown rows add each turning point with its type, the end behaviour in both directions, and the theoretical maximum number of roots and turns for comparison with what was actually found.
It is a different job from our polynomial arithmetic calculator, which adds, multiplies and evaluates polynomials symbolically, and from the polynomial division calculator, which performs long and synthetic division. Those manipulate the algebra. This page shows you the shape.
How to Use It
- Enter the coefficients highest power first. Include a zero for every missing power — x³ − 4x is "1, 0, −4, 0", not "1, −4, 0".
- Leave the window on automatic. It is set from the roots and turning points the tool finds, which is almost always the view you want.
- Override it when you need a specific region. Switching to manual lets you zoom into a particular interval to inspect behaviour near one root.
- Read the marked points. Roots are marked on the x-axis, turning points on the curve, and the y-intercept where the curve meets the vertical axis.
- Check a value by hand. Enter any x in the evaluate field and compare the tool's answer with your own substitution.
How the Roots and Turning Points Are Found
Every real root of a polynomial with real coefficients lies inside a bound that can be computed from the coefficients alone. Cauchy's bound says every root satisfies |x| ≤ 1 + max|ai ÷ an|, where an is the leading coefficient. That gives the tool a finite interval it knows is enough to search, rather than a guess. The general theory of where polynomial zeros can sit is set out in the DLMF section on the zeros of polynomials.
Inside that interval the tool samples finely and watches for a change of sign. A sign change guarantees a root in between, by the intermediate value theorem, and bisection then narrows the bracket until the root is located to about twelve significant figures. Repeating the process on the derivative gives the turning points, since a smooth curve turns exactly where its slope is zero.
Take the default, x³ − 6x² + 11x − 6. It factors as (x − 1)(x − 2)(x − 3), so the roots are 1, 2 and 3 and the tool finds all three. The derivative is 3x² − 12x + 11, whose roots are 2 ± √12 ÷ 6, giving turning points at about 1.4226 and 2.5774. Substituting x = 0 gives the y-intercept of −6, which is always simply the constant term.
The One Case Sign Changes Cannot Catch
Bracketing by sign change finds every root the curve crosses, and misses every root it merely touches. A repeated root of even multiplicity — the double root in (x − 2)², or the quadruple root in (x − 1)⁴ — is a point where the curve reaches the axis and turns back without ever changing sign. No bracket exists, so no bisection can find it.
The tool handles this by checking the turning points it has already located: wherever the polynomial and its derivative are both effectively zero at the same x, that point is reported as a repeated root. The quartic preset loads x⁴ − 4x³ + 4x², which is x²(x − 2)² and has double roots at 0 and 2, so you can see the curve sitting tangent to the axis at both.
The multiplicity also determines how the curve behaves at a root, and this is worth knowing without a calculator. An odd multiplicity means the curve crosses — multiplicity 1 crosses briskly, multiplicity 3 crosses with a flat inflection. An even multiplicity means the curve touches and turns back. That single rule lets you sketch the shape of a factored polynomial from its factors alone. Our rational root calculator finds the exact rational roots when they exist, and the quadratic equation solver handles the degree-2 case exactly.
End Behaviour Follows Two Facts Only
What a polynomial does at the far ends of the x-axis depends on exactly two things: the degree's parity and the leading coefficient's sign. Nothing else in the polynomial matters, because for large |x| the leading term dominates every other term put together.
Even degree with a positive leading coefficient rises at both ends. Even degree with a negative one falls at both ends. Odd degree with a positive leading coefficient falls on the left and rises on the right. Odd degree with a negative one does the reverse. That produces a useful corollary: every odd-degree polynomial has at least one real root, because it goes from negative infinity to positive infinity and must cross zero on the way. Even-degree polynomials carry no such guarantee, which is why the "no real roots" preset above is a quadratic.
The tool reports both ends explicitly, because reading end behaviour off a plotted window is unreliable — the window only ever shows a finite piece of the curve, and a polynomial that appears to be levelling off inside the window is always, eventually, heading for infinity.
Why the Viewing Window Is the Hard Part
A polynomial's interesting behaviour occupies a bounded region, and outside it the curve simply climbs or falls away. If the window is too wide, that bounded region compresses into a vertical line and you see nothing. If it is too narrow, you see a fragment and may miss roots entirely.
The automatic setting solves this by choosing the window from the features rather than from a default. It takes the smallest and largest of every real root and every turning point, spreads a margin of about 25 percent of that span on each side, and then sets the vertical range from the polynomial's values at those same feature points. The result is a view in which every root and every turn is visible with room to spare.
The vertical range needs its own care, because polynomial values grow far faster than x does. A degree-6 polynomial evaluated a little outside its root cluster can easily be in the millions, and letting that value set the top of the window flattens everything else onto the axis. The tool therefore takes the vertical range from the values at the turning points, not from the extremes at the window edges, and clips the curve where it leaves the box rather than rescaling to fit it.
Reading a Polynomial Graph Correctly
The roots are where the curve meets the x-axis, and they are the solutions of p(x) = 0 — not of p(x) = anything else. To solve p(x) = 7, subtract 7 from the constant term and look for the roots of the new polynomial. This is a routine substitution and it catches people out surprisingly often.
Turning points are where the derivative is zero, and they are local maxima or minima, not necessarily global ones. A cubic's local maximum can sit well below values the curve takes further to the right; the labels "maximum" and "minimum" are always local unless the polynomial's end behaviour makes them global, which for even degree with a positive leading coefficient it does for the lowest minimum.
Between turning points the curve is monotonic — strictly rising or strictly falling — which is why the count of turning points bounds the count of roots. A polynomial with no turning points at all is strictly monotonic and can have at most one real root. The connection between derivatives, critical points and the shape of a curve is developed in MIT's Single Variable Calculus course. If your coefficients came from fitting data rather than from algebra, our polynomial regression calculator is the page that produces them.
Arb Digital publishes hundreds of free calculators covering algebra, calculus, geometry and statistics — all free to use, with no sign-up and no limit on how often you run them.
Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Omitting zero coefficients. Every power from the highest down to the constant needs a slot, or you are plotting a different polynomial.
- Trusting a fixed viewing window. Roots outside the window are invisible, and nothing on the screen warns you they exist.
- Assuming every root shows as a crossing. Roots of even multiplicity touch the axis and turn back without changing sign.
- Reading end behaviour off the plot. The window shows a finite piece. End behaviour comes from the degree's parity and the leading sign, and from nothing else.
- Calling a local maximum the maximum. Turning points are local. An odd-degree polynomial has no global maximum or minimum at all.
Related Free Tools From Arb Digital
Use the polynomial arithmetic calculator to build or expand the polynomial, the polynomial division calculator to divide out a known factor, the rational root calculator to find exact rational roots, the quadratic equation solver for the degree-2 case, and the polynomial regression calculator when the coefficients come from data. More live in the free tools hub.
Frequently Asked Questions
A polynomial of degree n has exactly n roots counted with multiplicity over the complex numbers, but at most n of them are real and there may be far fewer. Complex roots of a real polynomial always come in conjugate pairs, so a real polynomial of odd degree always has at least one real root.
At most one fewer than its degree, because turning points are the roots of the derivative and the derivative has degree one lower. A cubic has at most two turning points, a quartic at most three, and either can have none at all.
Roots of even multiplicity touch the x-axis without crossing it, so there is no change of sign to bracket. This tool catches them separately by looking for points where the polynomial and its derivative are both zero, which is exactly what a repeated root is.
Only two things: whether the degree is odd or even, and the sign of the leading coefficient. For large values of x the leading term dominates everything else, so no other coefficient has any effect on what happens at the far ends.
Substitute x equals zero, which leaves only the constant term. The y-intercept of a polynomial is therefore always its constant coefficient, and it is the one feature you can read straight off the equation with no work at all.
It finds every real root and every turning point first, takes the smallest and largest of those, and adds a margin of about a quarter of the span at each end. The vertical range comes from the polynomial's values at the turning points rather than at the window edges, which stops one large value flattening everything else.
A root of even multiplicity touches the axis and turns back without crossing. A root of odd multiplicity greater than one crosses, but flattens as it does so. A simple root crosses briskly. The multiplicity is readable from the shape at the crossing.
Subtract that number from the constant term and find the roots of the resulting polynomial. Solving p(x) equals seven is the same as finding where p(x) minus seven crosses the axis, so only the constant coefficient changes.
This plotter is provided for study and for checking your own working. It is not a substitute for showing the algebraic steps your course or assessment requires.
