Step by step
The question
Question: find the end behavior of f(x) = 2x² − 5x + 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.
The biggest power of x here is x².
So the leading term is 2x².
2 Read the ends
The power 2 is even.
So both ends go the same way.
The number in front is positive, so the right end goes up.
Careful!
An even power makes a negative x positive.
So both ends do the same.
Far to the right, x gets bigger and bigger.
2x² 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.
2x² goes up: f(x) goes to infinity.
We can see it on a small sketch of 2x².
Left end up, right end up.
3 Check
Let's check with a big x: x = 10 in 2x² gives 200: big and positive, so up.
And x = −10: 200, a big positive number: up.
It matches!
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.