Characteristics of polynomials

		Odd-degree	Even-degree
Domain		$\{x\in\mathbb{R}\}$	$\{y\in\mathbb{R}\}$
Range	$+$ve leading coefficient	$\{x\in\mathbb{R}\}$	$\{y\in\mathbb{R}\mid y\ge k$ where $k\in\mathbb{R}\}$ $k$ is the absolute minimum
Range	$-$ve leading coefficient	$\{x\in\mathbb{R}\}$	$\{y\in\mathbb{R}\mid y\le k$ where $k\in\mathbb{R}\}$ $k$ is the absolute maximum
End behaviour	$+$ve leading coefficient	$Odd-degree polynomial with positive leading coefficient$ $y\to-\infty$ as $x\to-\infty$ $y\to+\infty$ as $x\to+\infty$ From QIII to QI	$Even-degree polynomial with positive leading coefficient$ $y\to+\infty$ as $x\to-\infty$ $y\to+\infty$ as $x\to+\infty$ From QII to QI
End behaviour	$-$ve leading coefficient	$Odd-degree polynomial with negative leading coefficient$ $y\to+\infty$ as $x\to-\infty$ $y\to-\infty$ as $x\to+\infty$ From QII to QIV	$Even-degree polynomial with negative leading coefficient$ $y\to-\infty$ as $x\to-\infty$ $y\to-\infty$ as $x\to+\infty$ From QIII to QIV
Symmetry		May be odd (but not always) Never even	May be even (but not always) Never odd
Number of real roots^*	Minimum	$1$	$0$
Number of real roots^*	Maximum	$n$	$n$
Number of turning points^*	Minimum	$0$ (always even)	$1$ (always odd)
Number of turning points^*	Maximum	$n-1$	$n-1$

^*$n$ is the degree of the polynomial.

The graph above shows the maximum number of real roots and turning points for the polynomial. Move the slider to change the degree of the polynomial. Use the check boxes to switch the sign of the leading coefficient, show or hide the roots and the turning points.

		Odd-degree	Even-degree
Domain		\(\{x\in\mathbb{R}\}\)	\(\{y\in\mathbb{R}\}\)
Range	\(+\)ve leading coefficient	\(\{x\in\mathbb{R}\}\)	\(\{y\in\mathbb{R}\mid y\ge k\) where \(k\in\mathbb{R}\}\) \(k\) is the absolute minimum
Range	\(-\)ve leading coefficient	\(\{x\in\mathbb{R}\}\)	\(\{y\in\mathbb{R}\mid y\le k\) where \(k\in\mathbb{R}\}\) \(k\) is the absolute maximum
End behaviour	\(+\)ve leading coefficient	$Odd-degree polynomial with positive leading coefficient$ \(y\to-\infty\) as \(x\to-\infty\) \(y\to+\infty\) as \(x\to+\infty\) From QIII to QI	$Even-degree polynomial with positive leading coefficient$ \(y\to+\infty\) as \(x\to-\infty\) \(y\to+\infty\) as \(x\to+\infty\) From QII to QI
End behaviour	\(-\)ve leading coefficient	$Odd-degree polynomial with negative leading coefficient$ \(y\to+\infty\) as \(x\to-\infty\) \(y\to-\infty\) as \(x\to+\infty\) From QII to QIV	$Even-degree polynomial with negative leading coefficient$ \(y\to-\infty\) as \(x\to-\infty\) \(y\to-\infty\) as \(x\to+\infty\) From QIII to QIV
Symmetry		May be odd (but not always) Never even	May be even (but not always) Never odd
Number of real roots^*	Minimum	\(1\)	\(0\)
Number of real roots^*	Maximum	\(n\)	\(n\)
Number of turning points^*	Minimum	\(0\) (always even)	\(1\) (always odd)
Number of turning points^*	Maximum	\(n-1\)	\(n-1\)