Skill: Predict end behaviour from degree and leading coefficient

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

1. The degree: Is it even or odd?
2. 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

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