Step by step
The question
Question: find the end behavior of f(x) = 4 + x − 2x³.
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.
Careful!
The leading term is not just the first part written.
It is the biggest power: −2x³.
The biggest power of x here is x³.
So the leading term is −2x³.
2 Read the ends
The power 3 is odd.
So the two ends go opposite ways.
The number in front is negative, so the right end goes down.
Far to the right, x gets bigger and bigger.
−2x³ goes down for ever.
Books write: as x goes to infinity, f(x) goes to minus infinity.
Far to the left, x is a big negative number.
−2x³ goes up: f(x) goes to infinity.
We can see it on a small sketch of −2x³.
Left end up, right end down.
3 Check
Let's check with a big x: x = 10 in −2x³ gives −2000: big and negative, so down.
And x = −10: 2000, a big positive number: up.
It matches!
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.