Quick questions on Graphing polynomials: leading-term behaviour, intercepts and root multiplicity

For large $|x|$, a polynomial behaves like its leading term $a_n x^n$.

If $(x - a)^m$ is a factor of $P(x)$, then $a$ is a root of multiplicity $m$.

Set $x = 0$ to find the y-intercept, which is the constant term of the polynomial.

A polynomial of degree $n$ has at most $n - 1$ turning points (where the derivative changes sign) and at most $n - 2$ points of inflection (where $P''$ changes sign). For Extension 1 sketches, you do not need to compute these precisely. You combine end behaviour with the root analysis to draw a continuous curve that hits the intercepts in the right way.

1. Sketch the end behaviour as two arrows on the axes. 2. Mark every x-intercept and label its multiplicity.

End behaviour: degree $2$, leading coefficient negative, both ends go to $-\infty$.

Sketch $P(x) = (x + 1)^2 (x - 2)(x - 4)$.

End behaviour: cubic with positive leading coefficient.

The graph does not cross the x-axis at $(x - 2)^2$; it touches and bounces back.

This is the single easiest sanity check; if your sketch passes through the y-axis at the wrong sign, the whole sketch is wrong.

A degree-$n$ polynomial has at most $n - 1$ turning points. A cubic can have at most one local max and one local min, never more.

Both cross the x-axis, but multiplicity 3 has a horizontal tangent at the crossing (flatter), like $x^3$ at the origin. :::