Polynomials & Rationals
A polynomial is completely determined by where it crosses the x-axis (its roots) and one stretch number. So skip the algebra: grab the roots and drag them. For Grade 12 (Ontario MHF4U and equivalents).
Build it from the roots
Every root r contributes a factor (x − r). Multiply the factors, scale by a leading coefficient a, and the whole curve follows:
Degree and the sign of a fix what happens far away — the famous end behaviour. Odd degree: the two ends go opposite ways. Even degree: both ends go the same way. And in between, a degree-n polynomial can turn around at most n − 1 times.
Drag the roots 🔵
Drag the dots along the x-axis (they snap to half-units). Stack two roots on the same spot and watch the curve bounce instead of cross. Then switch to Rational mode.
What to watch
Cross or bounce
A single root crosses the axis; a doubled root touches and bounces (the factor is squared, so the sign can't flip). That one observation solves most "sketch this polynomial" exam questions.
Rational = polynomial ÷ polynomial
Denominator roots become vertical asymptotes — the curve blows up. Unless the numerator has the same root: then the factors cancel and leave a hole (the open circle). Compare the degrees to find the horizontal asymptote.
In Rational mode, put a numerator root exactly on top of a denominator root and watch the vertical asymptote collapse into a small open circle — a removable hole. Then check the horizontal asymptote rule: fewer roots on top → y = 0; same count → y = a; more on top → no horizontal asymptote at all.