Step by step
The question
Question: find the end behavior of f(x) = (x − 2)(x + 1)(x + 4).
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.
Multiply the x's of the brackets.
Each bracket gives one x.
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.