Concave Up: Definition, Function & Graph

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  • 0:04 What Is Concave Up?
  • 1:04 What Concave Up Means
  • 3:04 The Math Behind Concavity
  • 5:22 Lesson Summary
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Lesson Transcript
Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

Concavity describes the shape of a graph, or function. In this lesson, learn about how a concave up shape relates to acceleration and to the second derivative in calculus.

What Is Concave Up?

A section of a curve is concave up if the y-value grows at a faster and faster rate moving from left to right. If you looked at the curve from above, it would remind you of the inside of a bowl. Using calculus, you would find that the first derivative was increasing in that interval and the second derivative was greater than zero.

The opposite of concave up, in which the y-value decreases from left to right, is called concave down. Here's a graph of a function with both types of concave. The red regions are concave up while the blue regions are concave down. Note that the concave up sections aren't just the ones that slope upwards, but also where a downward slope begins to decrease. The opposite is true for the concave down sections. This gradual change in y-value gives each concave section its bowl-like shape.

Graph with concave up (red) and concave down (blue) regions
Graph with concave up and down regions

You can remember the difference between concave up and concave down by imagining them as a series of bowls. The concave up sections would catch rain water, while the concave down sections are upside down bowls and would not.

What Concave Up Means

So what does a concave up section of a graph usually tell us? Usually it's a sign of rapid growth or decline. Suppose that your Aunt Felicity is a cat collector. Anytime she sees a stray wandering down the road, she slides open the door of her minivan, sets out a bowl of milk to tempt the furry creature inside, and then takes the new cat back to her cat farm. Because she is adding new cats daily, and because those cats seem to have an uncanny ability to reproduce quickly with the unique formula she feeds them, she is quickly overrun by cats, and happy about it.

Aunt Felicity's cat population starts off by growing very quickly and that growth appears to speed up each year for the first five years. In the first year of collecting cats, she adds 90 new cats. By year three, those cats have multiplied to 596, and by the fifth year Aunt Felicity has 2500 cats. Those first five years aren't just about an increase in the number of cats, but an increase in the rate of gain or growth. Aunt Felicity gets more new cats each year than she got the previous year. The cat population curve is concave up for these five years, as shown in the red section on the curve.

Graph of Aunt Felicitys cats
Graph of increasing curve

Another way to think about concave up is to relate it to acceleration. If our equation represented a change in distance over time, then a concave up section on the curve would mean there was positive acceleration or increasing speed. In our example, Aunt Felicity is 'picking up speed' with respect to adding to her cat collection.

By contrast, look at the blue section of the curve, the time between the fifth and tenth year, and notice that the curve is now concave down. Aunt Felicity is losing steam. In fact, during each year after the fifth, Aunt Felicity gradually collects fewer and fewer cats. Maybe she is running out of food. Maybe the cats are feeling crowded and wandering off. Maybe she is just getting bored with cats and has decided to take up oil painting. While she is still adding to her cat collection, she is doing so at a gradually decreasing rate. This trend causes the curve to reverse direction.

The Math Behind Concavity

A graph is concave up if the slope of the line tangent to the graph is gradually increasing as you move from left to right. It also should be true that the slopes of line segments connecting any two points, also known as secant lines, would become gradually larger, at least within a small enough region. In other words, the graph is becoming steeper, it's concave up. In the graph of Aunt Felicity's cat farm, you can see that the orange line is steeper than the green line. The graph is concave up in that region.

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