Graph of every polynomial function approaches either $-\infty$ or $+\infty$ as x approaches $-\infty$ or $+\infty$. How the graph of a function looks can be determined without actually plotting it. One such technique is called the end behaviour of a polynomial function.
What is a Polynomial Function?
A polynomial function is the simplest, most commonly used, and most important mathematical function. These functions represent algebraic expressions with certain conditions. There are a number of polynomial functions.
Polynomial is made up of two words, poly, and nomial. “Poly” means many, and “nomial” means the term, and hence polynomials are “algebraic expressions with one or many terms“.
A polynomial function in standard form is: $f(x) = a_nx_n + a_{n-1}x_{n-1} + … + a_2x_2+ a_1x + a_0$. This algebraic expression is called a polynomial function in variable $x$.
where,
- $a_n$, $a_{n-1}$, … $a_0$ are real number constants
- $a_n$ can’t be equal to zero and is called the leading coefficient
- $n$ is a non-negative integer
- Each exponent of a variable in polynomial function should be a whole number
One of the important terms associated with polynomials is degree. The degree of the polynomial function is the highest power of the variable it is raised to.
For example in a polynomial function $f(x) = -7x^3 + 6x^2 + 11x – 19$, the highest exponent found is $3$ from $-7x^3$. This means that the degree of this particular polynomial is $3$.
What Is Meant By End Behaviour Of A Polynomial?
The end behavior of a function $f$ describes the behavior of the graph of the function at the “ends” of the $x$-axis. In other words, the end behavior of a function describes the trend of the graph if we look to the right end of the $x$-axis (as $x$ approaches $+\infty$) and to the left end of the $x$-axis (as $x$ approaches $-\infty$).
In the above figure, you can see that the graph of the polynomial function approaches $+\infty$ as $x$ approaches $-\infty$ and it approaches $-\infty$ as $x$ approaches $+\infty$.
How To Determine The End Behaviour Of a Polynomial Function?
Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term and sign of its coefficient in the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as $x$ gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of the highest degree.
The following table will be helpful in determining the end behaviour of a polynomial.
| Degree | Leading Coefficient | End behaviour of the Polynomial Function | Graph of the Polynomial Function |
| Even | Positive | $f(x) → +\infty$, as $x → -\infty$ $f(x) → +\infty$, as $x → +\infty$ | $f(x) = x^2$  |
| Even | Negative | $f(x) → -\infty$, as $x → -\infty$ $f(x) → -\infty$, as $x → +\infty$ | $f(x) = -x^2$ |
| Odd | Positive | $f(x) → -\infty$, as $x → -\infty$ $f(x) → +\infty$, as $x → +\infty$ | $f(x) = x^3$ |
| Odd | Negative | $f(x) → +\infty$, as $x → -\infty$ $f(x) → -\infty$, as $x → +\infty$ | $f(x) = -x^3$ |
Examples
Let’s determine the end behaviour of some functions.
Example 1
Determine the end behaviour of a polynomial function $f(x) = -7x^{5}+2x^{4}+18x^{3}-x^{2}+3x+18$
The degree of a polynomial function is $5$ (Odd)
The sign of the leading coefficient is $-ve$
End behaviour: $f(x) → +\infty$, as $x → -\infty$ and $f(x) → -\infty$, as $x → +\infty$
Example 2
Determine the end behaviour of a polynomial function $f(x)=2x^{4}-5x^{3}+x^{2}-1$
The degree of a polynomial function is $4$ (Even)
The sign of the leading coefficient is $+ve$
End behaviour: $f(x) → +\infty$, as $x → -\infty$ and $f(x) → +\infty$, as $x → +\infty$
Conclusion
You can visualize the graph of a polynomial function without actually plotting it on graph paper if you know the end behaviour of a polynomial function. The end behaviour of a polynomial function can be summarized as:
- Degree: Even, Leading Coefficient: Positive, $f(x) → +\infty$, as $x → -\infty$ and $f(x) → +\infty$, as $x → +\infty$
- Degree: Even, Leading Coefficient: Negative, $f(x) → -\infty$, as $x → -\infty$ and $f(x) → -\infty$, as $x → +\infty$
- Degree: Odd, Leading Coefficient: Positive, $f(x) → -\infty$, as $x → -\infty$ and $f(x) → +\infty$, as $x → +\infty$
- Degree: Odd, Leading Coefficient: Negative, $f(x) → +\infty$, as $x → -\infty$ and $f(x) → -\infty$, as $x → +\infty$
