A polynomial function is built from powers of $x$ with constant coefficients. Its graph is always smooth and continuous: no sharp corners, no jumps, no holes. The leading term alone determines how the graph behaves far to the left and far to the right, which is called the end behavior.
Quick Reference
| Property | Rule |
|---|---|
| General form | $P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0$, $a_n \neq 0$ |
| Even degree, $a_n > 0$ | Both ends rise to $+\infty$ |
| Even degree, $a_n < 0$ | Both ends fall to $-\infty$ |
| Odd degree, $a_n > 0$ | Left end falls, right end rises |
| Odd degree, $a_n < 0$ | Left end rises, right end falls |
| Intermediate Value Theorem | If $P(a)$ and $P(b)$ have opposite signs, $P$ has a root between $a$ and $b$ |
| Graph shape | Always smooth (no sharp corners) and continuous (no breaks) |
Polynomial Functions
A polynomial function of degree $n$ is a function of the form
$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$where $n$ is a nonnegative integer and the leading coefficient $a_n \neq 0$. The term $a_n x^n$ is the leading term, and $a_0$ is the constant term.
Consider $P(x) = 4x^5 - 3x^3 + 7x - 9$. This polynomial has degree 5, leading coefficient 4, and constant term $-9$.
Power Functions
The basic building blocks of polynomial graphs are monomials $f(x) = x^n$, called power functions. Their shape depends on whether $n$ is even or odd.
Even powers of $x$ ($n$ even): graphs resemble $f(x) = x^2$, are U-shaped, symmetric about the $y$-axis, and both ends rise.
Odd powers of $x$ ($n$ odd): graphs resemble $f(x) = x^3$, are S-shaped, symmetric about the origin, and fall on the left while rising on the right.
Case 1: $n$ is Even
When $n$ is even, $f(x) = x^n$ is an even function because $f(-x) = (-x)^n = x^n = f(x)$. Therefore the graph is symmetric about the $y$-axis.
End behavior:
$f(x) \to +\infty \quad \text{as } x \to +\infty \qquad \text{and} \qquad f(x) \to +\infty \quad \text{as } x \to -\infty.$The graph is U-shaped with its lowest point at the origin and rises to positive infinity on both sides.
Case 2: $n$ is Odd
When $n$ is odd, $f(x) = x^n$ is an odd function because $f(-x) = (-x)^n = -x^n = -f(x)$. The graph is symmetric about the origin.
End behavior:
$f(x) \to +\infty \quad \text{as } x \to +\infty \qquad \text{and} \qquad f(x) \to -\infty \quad \text{as } x \to -\infty.$The graph passes through the origin with an S-shape and becomes flatter near the origin and steeper farther away as $n$ increases.
Sketch graphs of the following functions.
(a) $P(x) = -x^4$
(b) $Q(x) = (x-3)^5 + 2$
Solution
(a) The graph of $P(x) = -x^4$ is a reflection of $y = x^4$ across the $x$-axis: a downward-opening U-shape with vertex at the origin. (b) The graph of $Q(x) = (x-3)^5 + 2$ is obtained from $y = x^5$ by shifting right 3 units and up 2 units. It has the same S-shape as $x^5$ but centered at $(3, 2)$ instead of the origin.Sketch graphs of the following functions.
(a) $P(x) = -2(x+1)^6 + 5$
(b) $Q(x) = 3(x-4)^7 - 10$
Solution
(a) Start with $y = x^6$ (even, U-shaped). Shift left 1 unit, reflect across the $x$-axis, stretch vertically by 2, then shift up 5 units. The result opens downward with vertex at $(-1, 5)$. (b) Start with $y = x^7$ (odd, S-shaped). Shift right 4 units, stretch vertically by 3, then shift down 10 units. The inflection point moves to $(4, -10)$.
End Behavior of Polynomial Functions
The end behavior of a polynomial is determined entirely by its leading term $a_n x^n$. When $|x|$ is large, all lower-degree terms become negligible:
$P(x) = x^n\!\left(a_n + \frac{a_{n-1}}{x} + \cdots + \frac{a_0}{x^n}\right) \approx a_n x^n.$The figures below illustrate this for $P(x) = -0.25x^3 + 10x^2 + 7x + 1$ compared to $f(x) = -0.25x^3$:
The four end-behavior rules are:
| Degree | Leading coeff | As $x \to +\infty$ | As $x \to -\infty$ |
|---|---|---|---|
| Even | $a_n > 0$ | $P(x) \to +\infty$ | $P(x) \to +\infty$ |
| Even | $a_n < 0$ | $P(x) \to -\infty$ | $P(x) \to -\infty$ |
| Odd | $a_n > 0$ | $P(x) \to +\infty$ | $P(x) \to -\infty$ |
| Odd | $a_n < 0$ | $P(x) \to -\infty$ | $P(x) \to +\infty$ |
Determine the end behavior of:
(a) $P(x) = -3x^6 + 5x^4 - 2x^2 + 7$
(b) $Q(x) = -\dfrac{1}{2}x^5 + 3x^4 + x - 1$
Solution
(a) The leading term is $-3x^6$: even degree, negative leading coefficient. Both ends fall: $P(x) \to -\infty \quad \text{as } x \to +\infty \qquad \text{and} \qquad P(x) \to -\infty \quad \text{as } x \to -\infty.$ (b) The leading term is $-\frac{1}{2}x^5$: odd degree, negative leading coefficient. The graph falls on the right and rises on the left: $Q(x) \to -\infty \quad \text{as } x \to +\infty \qquad \text{and} \qquad Q(x) \to +\infty \quad \text{as } x \to -\infty.$
Continuity of Polynomial Graphs
(Intermediate Value Theorem for Polynomials) If $k$ is any value between $P(a)$ and $P(b)$, then there exists at least one number $c$ between $a$ and $b$ such that $P(c) = k$.
Polynomial graphs are continuous: you can trace the entire graph without lifting your pen. Because of this, they satisfy the Intermediate Value Theorem. Think of driving a car: if your speedometer reads 45 mph at one moment and 65 mph later, it must have shown 50 mph at some point in between.
A Special Case: Locating Roots
If $P(a) \cdot P(b) < 0$ (that is, $P(a)$ and $P(b)$ have opposite signs), then $P(c) = 0$ for some $c$ between $a$ and $b$.
This is a practical tool: if we find two points where the polynomial has opposite signs, there must be a root between them. This is the basis of numerical root-finding methods such as the bisection method.
Every polynomial of odd degree has at least one real root, because its end behavior forces it to take on both positive and negative values, and by continuity it must cross zero.
Smoothness of Polynomial Graphs
Polynomial graphs are also smooth: they have no sharp corners or sudden direction changes. Every turn in the graph is gradual and rounded. This contrasts with functions like $|x|$, which has a sharp corner at the origin, or piecewise functions with abrupt changes.
