End behavior of functions & their graphs
About this video
The video uses three graphs to find the function that increases as x goes toward infinity and decreases as x goes toward negative infinity. h(x) fails the first condition, g(x) rises on both ends so it fails the second, and only f(x) meets both.
You'll be able to
- describe what a graph does as x goes toward positive or negative infinity.
- test each graph against two end-behaviour conditions.
- ignore local bumps and focus on the far ends of a graph.
Key words
- end behaviour
- What a function's values do at the far left and far right of its graph.
- infinity
- The idea of x getting larger and larger without stopping.
- local maximum / minimum
- A peak or valley that is highest or lowest only in its nearby area.
- global minimum
- The lowest point of the whole graph.
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.
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