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Week 3
Topic 4 – Concavity
Introduction
As you read from left to right, you can see how functions increase or decrease. However, that’s not the end of the story. An increasing function has a positive average rate of change, and a decreasing function has a negative average rate of change. Also affecting the shape of a graph is how the rate of change itself is changing. That sounds complicated, so we’re going to make it simple first.
Note
This is supplemental material not in the book, which means you don’t have any homework problems on it. Luckily for you, there are sample problems in the PCC Math 111 Supplement (in Desire2Learn, click Links to find it). You might see this on the first exam. You will definitely see concavity again in Calculus I.
Reading
Concavity comes in two types, up and down. This is a property we associate with x- intervals, so a graph might be concave up for a while, and then switch to concave down.
Let’s start with a couple straight lines, one increasing and one decreasing.
increasing line
-1 (^) -1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
x
y (^) decreasing line
-1 (^) -1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
x
y
A straight line is neither concave up nor concave down. But picture this line as a piece of wire, and bend this line up. Now the solid line in this graph is concave up.
increasing line concave up
-1 (^) -1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
x
y
decreasing line concave up -1 (^) -1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
x
y
Similarly, if we start with a straight line and bend it down a little bit, we say the graph is concave down.
increasing line concave down
-1 (^) -1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
x
y decreasing line concave down
-1 (^) -1 1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
x
y
Let’s spend a few minutes now examining concavity as the change in rates of change.
Here are a few graphs of f ( ) x = x^2 − 4 with secant lines drawn through various points. Notice this function is concave up on (-∞,∞).
f(x)=x^2- -5 -4 -3 -2 -1-4-2 (^1) Secant Line 1
2
4
6
8
10
12
14
16
x
y
f(x)=x^2- -5 -4 -3 -2 -1-4-2 (^1) Secant Line 2
2
4
6
8
10
12
14
16
x
y
f(x)=x^2- -5 -4 -3 -2 -1-4-2 (^1) Secant Line 3
2
4
6
8
10
12
14
16
x
y
Look at the slopes of the secant lines. Secant Line 1 has a slope of -5, Secant Line 2 has a slope of -1, and Secant Line 3 has a slope of 3. As we move from left to right, the slopes are increasing. The average rate of change is increasing. (Keep in mind that increasing can also mean “becoming less negative”, so the numbers -10, -8, and -6 are also increasing.)
Similarly, if a function’s slope is decreasing (or becoming more negative), then the function will be concave down. Try to sketch this out yourself by graphing a function like
f ( ) x = x and comparing the slopes of three secant lines. As the x-values increase, the secant line slopes will become smaller (closer to zero).
To summarize: When a function is concave up, its average rate of change is increasing. When a function is concave down, its average rate of change is decreasing.
These are more technical definitions than “bends up” or “bends down”, and they’re very important in Calculus^1 , but in this course you’re welcome to keep it casual.
Be careful not to mix up concavity with increasing/decreasing. They are separate concepts. The first page of this reading shows graphs that are increasing and concave up, increasing and concave down, decreasing and concave up, and decreasing and concave down.
(^1) This is a pre-calculus course, so I pretend that every reader will be studying Calculus soon. If you won’t be, you
can safely disregard the references.
Concept Exercises
Exercise 1: Here are the graphs of 6 functions.
a. Fill in the first blank with whether the function is increasing, decreasing, or constant b. Fill in the second blank with whether the function is concave up, concave down, or neither.
x
f(x)
____________________
____________________
x
f(x)
____________________
____________________
x
f(x)
____________________
____________________
x
f(x)
____________________
____________________
x
f(x)
____________________
____________________
x
f(x)
____________________
____________________
Exercise 2: Determine the interval(s) on which the functions graphed below are concave up or concave down.
a.
The graph of y = s(x)
b.
The graph of y = t(x)
