Graphs of polynomial functions

Sketch polynomials from their factored form using the leading term, the zeroes and the sign, and how the graph behaves at simple and multiple zeroes.

Behaviour for large x

Polynomial graphs are continuous and smooth. For large $|x|$ the leading term swamps all the others, so the leading term alone decides where the two ends of the graph go.

	Leading coefficient positive	Leading coefficient negative
Even degree	both ends up	both ends down
Odd degree	down on the left, up on the right	up on the left, down on the right

Simple and multiple zeroes

If $(x - a)^m$ is a factor of $P(x)$ but $(x - a)^{m+1}$ is not, then $a$ is a zero of multiplicity $m$. Multiplicity $1$ is a simple zero; multiplicity $2$ or more is a multiple (repeated) zero.

y = ¼(x + 2)ᵐ(x − 1)ⁿ

Change the multiplicities and watch the graph at each zero. The degree is m + n, which also decides where the ends go.

Try this

Which values of $m$ and $n$ make both ends of the graph point upward?

Sketching from factors

Example 1●●

Sketch $y = -(x + 1)^2(x - 3)$.

Leading term $-x^3$: up on the left, down on the right.

−x² × x = −x³.

Double zero at $x = -1$ (touches), simple zero at $x = 3$ (crosses).

$y$-intercept: $-(1)^2(-3) = 3$.

Put x = 0.

Coming down from the top left, it touches the axis at $-1$, rises through $(0, 3)$, then turns and crosses down through $3$.

Example 2●●●

Sketch $y = x^3(x - 2)^2$.

Degree $5$, leading coefficient positive: down on the left, up on the right.

Triple zero at $0$: crosses and flattens. Double zero at $2$: touches.

Negative for $x < 0$, positive for $0 < x < 2$ and for $x > 2$.

Flashcards

Ends of an even-degree polynomial

Both point the same way: up if the leading coefficient is positive.

Zero of odd multiplicity

The graph crosses the axis.

Zero of even multiplicity

The graph touches the axis.

Multiplicity 3

Crosses and flattens out (horizontal tangent).

Practice

As $x \to \infty$, $y = -2x^4 + x^3 + 5$:
At a zero of multiplicity $2$, the graph:
Find the $y$-intercept of $y = -(x + 1)^2(x - 3)$.
What is the multiplicity of the zero $x = 0$ of $y = x^3(x - 2)^2$?
Find the degree of $y = x^3(x - 2)^2$.
An odd-degree polynomial with a positive leading coefficient goes:
For $y = x^3(x - 2)^2$, where is $y < 0$?
The graph of $y = (x - 4)^3$ at $x = 4$:
How many distinct $x$-intercepts does $y = (x + 1)^2(x - 3)^3$ have?
Which equation could match a quartic that touches the axis at $x = 1$ and crosses at $x = -2$ and $x = 3$?