Skill: Determine end behaviour of a function

End behaviour describes what happens to $(f(x))$ when $x$ becomes very large positive or very large negative.

For a polynomial, the leading term determines the end behaviour. The leading term is the term with the highest power of $x$.

Worked example

Determine the end behaviour of

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

Step 1: Identify the leading term.

The leading term is

$$-2x^3.$$

The lower-degree terms become less important when $(|x|)$ is very large.

Step 2: Examine the right end, when $(x\to\infty)$.

As $(x\to\infty)$,

$$x^3\to\infty.$$

Because the coefficient is negative,

$$-2x^3\to-\infty.$$

Therefore,

$$\boxed{\lim_{x\to\infty}f(x)=-\infty}.$$

In words: As $x$ increases without bound, $(f(x))$ decreases without bound.

Step 3: Examine the left end, when $(x\to-\infty)$.

As $(x\to-\infty)$,

$$x^3\to-\infty.$$

Multiplying by $(-2)$ changes the direction:

$$-2x^3\to\infty.$$

Therefore,

$$\boxed{\lim_{x\to-\infty}f(x)=\infty}.$$

In words: As $x$ decreases without bound, $(f(x))$ increases without bound.

So the graph rises on the far left and falls on the far right.