Step by step
The question
Question: find the end behavior of f(x) = x³ − 4x + 1.
1 Find the leading term
The leading term is the part with the biggest power of x.
It decides where the graph goes at the far ends.
The biggest power of x here is x³.
So the leading term is x³.
2 Read the ends
The power 3 is odd.
So the two ends go opposite ways.
The number in front is positive, so the right end goes up.
Careful!
An odd power keeps the sign of x.
A big negative x gives the opposite of a big positive x.
Far to the right, x gets bigger and bigger.
x³ goes up for ever.
Books write: as x goes to infinity, f(x) goes to infinity.
Far to the left, x is a big negative number.
x³ goes down: f(x) goes to minus infinity.
We can see it on a small sketch of x³.
Left end down, right end up.
3 Check
Let's check with a big x: x = 10 in x³ gives 1000: big and positive, so up.
And x = −10: −1000, a big negative number: down.
It matches!
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.