Step by step
The question
Question: find the end behavior of f(x) = −x²(x − 1)(x + 3).
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, and the −1 in front.
Each bracket gives its x, to its power.
So the leading term is −x⁴.
2 Read the ends
The power 4 is even.
So both ends go the same way.
The number in front is negative, so the right end goes down.
Careful!
An even power makes a negative x positive.
So both ends do the same.
Far to the right, x gets bigger and bigger.
−x⁴ 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.
−x⁴ goes down: f(x) goes to minus infinity.
We can see it on a small sketch of −x⁴.
Left end down, right end down.
3 Check
Let's check with a big x: x = 10 in −x⁴ gives −10000: big and negative, so down.
And x = −10: −10000, a big negative number: down.
It matches!
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.