Predict end behaviour from degree and leading coefficient

The end behaviour of a polynomial $f(x)=a_nx^n+\cdots+a_0$ is determined by the parity of its degree and the sign of its leading coefficient: even degree gives matching tail directions, while odd degree gives opposite directions; a positive leading coefficient makes the right tail rise and a negative one makes it fall. This connects the algebraic structure of a polynomial to the graph’s shape as $x$ approaches positive or negative infinity, without requiring analysis of its individual zeros or turning points.

Detailed Explanation: Predict end behaviour from degree and leading coefficient

To predict a polynomial’s end behaviour, look only at:

The degree: Is it even or odd?
The leading coefficient: Is it positive or negative?

The leading term is the term with the highest power of $x$ .

Even degree: both ends point in the same direction.
Odd degree: the ends point in opposite directions.
Positive leading coefficient: the right end rises.
Negative leading coefficient: the right end falls.

Example

Predict the end behaviour of

f(x)=-2x^5+3x^3-x+7.

Step 1: Identify the degree.

The highest power of $x$ is $5$ , so the polynomial has degree $5$ .

Since $5$ is odd, the ends of the graph point in opposite directions.

Step 2: Identify the leading coefficient.

The leading term is $-2x^5$ , so the leading coefficient is $-2$ .

Since $-2$ is negative, the graph’s right end falls. In symbols,

\text{as } x\to+\infty,\quad f(x)\to-\infty.

Step 3: Determine the left end.

Because the degree is odd, the left end must point in the opposite direction from the right end. Therefore, the left end rises:

\text{as } x\to-\infty,\quad f(x)\to+\infty.

So the end behaviour is

\boxed{\text{left end rises and right end falls}.}

Learn by doing: Predict end behaviour from degree and leading coefficient

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Function End Behaviour (Polynomials) - Rule to Behaviour

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