Step by step
The question
Question: find the end behavior of f(x) = −3x³ + x² − 7.
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 −3x³.
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.
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.
−3x³ 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.
−3x³ goes up: f(x) goes to infinity.
We can see it on a small sketch of −3x³.
Left end up, right end down.
3 Check
Let's check with a big x: x = 10 in −3x³ gives −3000: big and negative, so down.
And x = −10: 3000, a big positive number: up.
It matches!
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.