Sketching a polynomial needs a handful of key features: where it crosses the axes, where it turns and how it behaves at the far left and right. These follow from the polynomial and its first two derivatives.
How to use the Polynomial Graphing Calculator
- Polynomial degree
- a₆ (x⁶)
- a₅ (x⁵)
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀
- Evaluate P(x) at x =
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How it works
Roots are found numerically and polished with Newton’s method. Turning points are classified with the second derivative. End behaviour depends on the degree (even or odd) and the sign of the leading coefficient.
Formula
Zeros: P(x) = 0 · Critical points: P′(x) = 0 · Inflection points: P″(x) = 0
Worked example
Inputs — Polynomial degree: 3; a₃ (x³): 1; a₂ (x²): 0; a₁ (x): -3; a₀: 1; Evaluate P(x) at x =: 2.
Results — Real zeros = x = −1.879385242, x = 0.3472963553, x = 1.532088886; Turning points = (−1, 3) local maximum
(1, −1) local minimum; Inflection points = (0, 1); y-intercept = 1; End behaviour = As x → −∞, y → −∞; as x → +∞, y → +∞; Value at x = P(2) = 3.
Key terms
- Critical point
- A point where the derivative is zero.
- Inflection point
- A point where concavity changes.
Frequently asked questions
How many turning points can a polynomial have?
At most n − 1 for degree n.
How do I find the end behaviour?
Look at the leading term: its sign and whether the degree is even or odd.
Why might a cubic have no turning points?
If P′(x) has no real roots, e.g. x³ + x.
What is an inflection point?
Where the curve changes from concave up to concave down.