Step by step
The question
Question: find the end behavior of f(x) = −x⁴ + 3x².
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 x⁴.
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.