Amy has a master's degree in secondary education and has taught math at a public charter high school.
There are all kinds of graphs, and we have various ways of describing our curves. We have words such as monotonic, concave up, and concave down. Watch this video lesson to learn how to identify these kinds of graphs.
Most functions will graph into a curve of some kind. Of course, some of these curves don't curve at all, while other curves look like a very serious roller coaster. For example, the function y = 3x graphs into a straight line with no curves at all, but the function y = 4 sin x graphs into a pretty cool roller coaster.
Look at all the steep dips this function takes. And the ride up the hills are pretty steep, too. I can only imagine the speed that a roller coaster shaped like this function can reach as it speeds down into the valleys.
A Monotonic Curve
Now, in math, as we look at the shapes of these graphs, we have terms that describe how the graph twists and bends. We say a a graph is monotonic if the graph never decreases or never increases. We can say that our monotonic graphs either decrease (non-increasing) or increase (non-decreasing).
For example, the graph of y = 3x
is monotonic because it is continually increasing. It never decreases.
Look at the graph of y = (x^3) / 2. Do you think this graph is monotonic?
If you said yes, then you are absolutely correct! This graph, although it has a flat part around x = 0, is still monotonic because it never goes down. It goes up, flattens out, and then goes up again. It's also increasing.
You can also double check that a function is monotonic by looking at its first derivative. If the first derivative never changes sign (i.e. going from positive to negative or negative to positive), then the function is monotonic. For example, if we take the derivative of y = 3x, we get y' = 3. This function does not change sign as we would expect from a monotonic function. It stays positive and never changes to negative. On the other hand, the function y = 4 sin x has a first derivative of y' = 4 cos x. Graphing this function you see that it changes sign quite frequently. It keeps going from positive to negative and back again. Therefore, this function is not a monotonic function.
If a graph has several curves, we can also analyze each of those curves. A curve is concave up if it is a curve that dips down and up again. It will look like a valley. This is the part of the roller coaster where you go really fast down to the bottom and then you go back up.
For the graph of the function y = 4 sin x, all the valleys are concave up curves.
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All the highlighted parts are where the graph is concave up. Note that the concave up parts don't start until the graph starts to dip down. Even though the graph is going down before the concave up parts, the graph is actually coming down from a hill. It needs to finish coming down before it dips further into the concave up part.
Mathematically, we can check whether a function is concave up at a particular point c by finding the second derivative and evaluating at our point c to see if it is positive. Let's try this with the function y = 4 sin x. The second derivative is y'' = -4 sin x. Evaluating this at the point x = -1, we get y'' = -4 sin (-1) = 0.0698. It's a positive number, so our function is concave up at this point. As you can see on the graph, our curve at x = -1 is inside the highlighted concave up area.
When our function's curve goes up and then down again, we have a concave down part. Here are the concave down parts of our graph y = 4 sin x.
In these regions, our second derivative is negative. Let's check it. Our second derivative is y'' = -4 sin x. Plugging in a 1 for x, we get y'' = -4 sin (1) = -.0698. This is a negative number telling us the curve is concave down at this point. Is this point inside the highlighted concave down parts? Yes, it is!
Let's review what we've learned now. We say a a graph is monotonic if the graph never decreases or never increases. A non-decreasing monotonic graph can also be referred to as increasing. A non-increasing monotonic graph can also be referred to as decreasing. A curve in a graph is concave up if it is a curve that dips down and up again. The curve in the graph is concave down if the curve goes up and then down again. Monotonic functions have first derivatives that don't change positive and negative signs. Concave up curves have second derivatives that are positive at those points. Concave down curves have second derivatives that are negative at those points.
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