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## Algebra (all content)

### Course: Algebra (all content) > Unit 10

Lesson 34: End behavior of polynomial functions# End behavior of functions & their graphs

Sal picks a function that has a given end behavior based on its graph. Created by Sal Khan.

## Want to join the conversation?

- @1:40Can you have several local Maximum and minimum points in a function?(7 votes)
- Yes. There are some videos on the site about local minimum and maximum points. Local minimums and maximums happen when you have wavy-looking functions. The bottoms and tops of the waves would all be local minimums and maximums.(15 votes)

- at1:42, what is a "local minimum/maximum" point versus a "global minimum/maximum" point?(7 votes)
- A local minimum is a point where changing
`x`

by a tiny amount in either direction causes`f(x)`

to increase. However, there can be an unlimited number of local minimums, and in between each pair of local minimums there is a local maximum. However there can be only one global minimum, which is the uniquely lowest point the function ever returns.(4 votes)

- f(x) does seem to be increasing as x increases, but at any point will the graph become vertical?(3 votes)
- At1:35Sal mention local minimum and maximum points and later global maximum points. What do these mean?

Did he do a video on it? Can you please explain to me what they mean by local and global and maximum and minimum points?

Thanks.(2 votes) - Is trigonometry harder or precalculus? (Just curious. I moved to US not long ago.) :)(1 vote)
- Trig is a subset of precalculus, so precalculus would be harder by definition.(3 votes)

- How can you tell witch way the lines are moving or do you not need to know that?(2 votes)
- You can say that the lines move both ways. The graphs shown are all continuous and have domains of all reals. In other words, any x value, no matter how large or small, can be put into the functions and a y value can be found.(1 vote)

- How do we calculate the turning point exactly?

Example: (0,9)(2 votes) - I came across a question asking me if a function was periodic.

What does this mean?(2 votes)- Function f(x) is periodic if and only if:

f(x + P) = f(x)

Where P is a nonzero constant (commonly referred to as the fundamental period).

A periodic function is basically a function that repeats after certain gap like waves. For example, the cosine and sine functions (i.e. f(x) = cos(x) and f(x) = sin(x)) are both periodic since their graph is wavelike and it repeats. On the other hand, f(x) = x (the parent linear function) graphs a simple line and there is no evident repeating pattern in its graph and upon analyzing the domain of the function we see that it does not satisfy the property f(x + P) = f(x). Therefore we say this function is aperiodic.(1 vote)

- Also can there be more than one local minimum and maximum local points in one function?(1 vote)
- Yes, you can have any number of local max or min in a function (though for polynomials there are set numbers of each that you can have).(2 votes)

- To clarify, I have the function f(x)=2/x^2+4x+3, the end behavior is when x approaches really low and really high values then my function gets closer to zero(1 vote)

## Video transcript

Which function increases as
x increases toward infinity and decreases as x decreases
toward negative infinity? So let's think about each
of these constraints. So, first, which function
increases as x increases? So as x increases
toward infinity. So x is going in that direction. So first let's look
at f of x over here. So f of x, as we get beyond this
minimum point right over here, as we increase our x, f of
x seems to be increasing. So f of x seems to make
the first constraint. Now let's think about g of x. Once we get past this minimum
point right over here, as x gets larger and
larger and larger, as it approaches
infinity, g of x seems to be getting larger
and larger and larger. g of x is moving up. So g of x also seems to
make this first constraint. Now let's think about h of x. As x moves towards
infinity, as x moves towards positive infinity,
h of x seems to be decreasing. So h of x does not even
make the first constraint. So our only two possibilities
are now g of x and f of x. So which of these
decrease as x decreases toward negative infinity? So let's think about
that, x decreasing toward negative infinity. We're going to be going
in that direction. So first let's look at f of x. So f of x, it kind of
goes up and down here. But after we hit this
little local maximum point-- this was a local
minimum point over here, not at a global one--
as we move to the left of this local maximum point,
as we get smaller and smaller x's, we see that the
function is decreasing. So it does seem to meet
the second constraint. It decreases as x decreases
toward negative infinity. So it meets that constraint. Now, what about g of x? After we have this
minimum point-- and actually it looks like
a global minimum point-- after we hit this minimum point
right over here, as x decreases toward negative infinity, g
of x seems to be increasing, not decreasing. So g of x does not meet
the second constraint. So the only function that
met both constraints seems to be f of x.